二项式系数 | DP-9

以下是二项式系数的常见定义。

A binomial coefficient C(n, k) can be defined as the coefficient of x^k in the expansion of (1 + x)^n.

A binomial coefficient C(n, k) also gives the number of ways, disregarding order, that k objects can be chosen from among n objects more formally, the number of k-element subsets (or k-combinations) of a n-element set.

问题
编写一个函数，它接受两个参数 n 和 k，并返回二项式系数 C(n, k) 的值。例如，你的函数应该返回6对于n = 4且k = 2，并且应当对于n = 5且k = 2返回10。

1) 最优子结构
C(n, k) 的值可以使用以下二项式系数的标准公式递归计算。

C(n, k) = C(n-1, k-1) + C(n-1, k)
   C(n, 0) = C(n, n) = 1

下面是一个简单的递归实现，它简单地遵循上面提到的递归结构。

C++

// A naive recursive C++ implementation
#include 
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    // Base Cases
    if (k > n)
        return 0;
    if (k == 0 || k == n)
        return 1;
 
    // Recur
    return binomialCoeff(n - 1, k - 1)
           + binomialCoeff(n - 1, k);
}
 
/* Driver code*/
int main()
{
    int n = 5, k = 2;
    cout << "Value of C(" << n << ", " << k << ") is "
         << binomialCoeff(n, k);
    return 0;
}
 
// This is code is contributed by rathbhupendra

C

// A Naive Recursive Implementation
#include 
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    // Base Cases
    if (k > n)
        return 0;
    if (k == 0 || k == n)
        return 1;
 
    // Recur
    return binomialCoeff(n - 1, k - 1)
           + binomialCoeff(n - 1, k);
}
 
/* Driver program to test above function*/
int main()
{
    int n = 5, k = 2;
    printf("Value of C(%d, %d) is %d ", n, k,
           binomialCoeff(n, k));
    return 0;
}

Java

// JAVA Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
import java.util.*;
 
class GFG {
 
    // Returns value of Binomial
    // Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
 
        // Base Cases
        if (k > n)
            return 0;
        if (k == 0 || k == n)
            return 1;
 
        // Recur
        return binomialCoeff(n - 1, k - 1)
            + binomialCoeff(n - 1, k);
    }
 
    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int n = 5, k = 2;
        System.out.printf("Value of C(%d, %d) is %d ", n, k,
                          binomialCoeff(n, k));
    }
}
 
// This code is contributed by Arnav Kr. Mandal.

Python

# A naive recursive Python implementation
 
 
def binomialCoeff(n, k):
 
    if k > n:
        return 0
    if k == 0 or k == n:
        return 1
 
    # Recursive Call
    return binomialCoeff(n-1, k-1) + binomialCoeff(n-1, k)
 
 
# Driver Program to test ht above function
n = 5
k = 2
print "Value of C(%d,%d) is (%d)" % (n, k,
                                     binomialCoeff(n, k))
 
# This code is contributed by Nikhil Kumar Singh (nickzuck_007)

C#

// C# Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
using System;
 
class GFG {
 
    // Returns value of Binomial
    // Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
 
        // Base Cases
        if (k > n)
            return 0;
        if (k == 0 || k == n)
            return 1;
 
        // Recur
        return binomialCoeff(n - 1, k - 1)
            + binomialCoeff(n - 1, k);
    }
 
    /* Driver program to test above function */
    public static void Main()
    {
        int n = 5, k = 2;
        Console.Write("Value of C(" + n + "," + k + ") is "
                      + binomialCoeff(n, k));
    }
}
 
// This code is contributed by Sam007.

PHP

 $n)
        return 0;
    if ($k==0 || $k==$n)
        return 1;
     
    // Recur
    return binomialCoeff($n - 1, $k - 1) +
               binomialCoeff($n - 1, $k);
}
 
    // Driver Code
    $n = 5;
    $k = 2;
    echo "Value of C","(",$n ,$k,") is "
               , binomialCoeff($n, $k);
 
// This code is contributed by aj_36
?>

Javascript

// javascript Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
 
   // Returns value of Binomial
// Coefficient C(n, k)
function binomialCoeff(n , k)
{
 
    // Base Cases
    if (k > n)
        return 0;
    if (k == 0 || k == n)
        return 1;
 
    // Recur
    return binomialCoeff(n - 1, k - 1)
        + binomialCoeff(n - 1, k);
}
 
/* Driver program to test above function */
var n = 5, k = 2;
document.write("Value of C("+n+", "+k+") is "+binomialCoeff(n, k));
 
// This code is contributed by Amit Katiyar

输出

Value of C(5, 2) is 10

2) 重叠子问题
需要注意的是，上述函数一次又一次地计算相同的子问题。对于 n = 5 和 k = 2，请参见以下递归树。函数C(3, 1) 被调用两次。对于较大的 n 值，将有许多常见的子问题。

C(5,2) 的二项式系数递归树

由于再次调用相同的子问题，该问题具有重叠子问题的性质。所以二项式系数问题具有动态规划问题的两个属性(见this和this)。与其他典型的动态规划 (DP) 问题一样，可以通过以自底向上的方式构造临时二维数组 C[][] 来避免对相同子问题的重新计算。以下是基于动态规划的实现。

C++

// A Dynamic Programming based solution that uses
// table C[][] to calculate the Binomial Coefficient
#include 
using namespace std;
 
// Prototype of a utility function that
// returns minimum of two integers
int min(int a, int b);
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    int C[n + 1][k + 1];
    int i, j;
 
    // Calculate value of Binomial Coefficient
    // in bottom up manner
    for (i = 0; i <= n; i++) {
        for (j = 0; j <= min(i, k); j++) {
            // Base Cases
            if (j == 0 || j == i)
                C[i][j] = 1;
 
            // Calculate value using previously
            // stored values
            else
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
        }
    }
 
    return C[n][k];
}
 
// A utility function to return
// minimum of two integers
int min(int a, int b) { return (a < b) ? a : b; }
 
// Driver Code
int main()
{
    int n = 5, k = 2;
    cout << "Value of C[" << n << "][" << k << "] is "
         << binomialCoeff(n, k);
}
 
// This code is contributed by Shivi_Aggarwal

C

// A Dynamic Programming based solution
// that uses table C[][] to
// calculate the Binomial Coefficient
#include 
 
// Prototype of a utility function that
// returns minimum of two integers
int min(int a, int b);
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    int C[n + 1][k + 1];
    int i, j;
 
    // Calculate value of Binomial Coefficient
    // in bottom up manner
    for (i = 0; i <= n; i++) {
        for (j = 0; j <= min(i, k); j++) {
            // Base Cases
            if (j == 0 || j == i)
                C[i][j] = 1;
 
            // Calculate value using
            // previously stored values
            else
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
        }
    }
 
    return C[n][k];
}
 
// A utility function to return
// minimum of two integers
int min(int a, int b) { return (a < b) ? a : b; }
 
/* Drier program to test above function*/
int main()
{
    int n = 5, k = 2;
    printf("Value of C(%d, %d) is %d ", n, k,
           binomialCoeff(n, k));
    return 0;
}

Java

// A Dynamic Programming based
// solution that uses table C[][] to
// calculate the Binomial Coefficient
 
class BinomialCoefficient {
    // Returns value of Binomial
    // Coefficient C(n, k)
    static int binomialCoeff(int n, int k)
    {
        int C[][] = new int[n + 1][k + 1];
        int i, j;
 
        // Calculate  value of Binomial
        // Coefficient in bottom up manner
        for (i = 0; i <= n; i++) {
            for (j = 0; j <= min(i, k); j++) {
                // Base Cases
                if (j == 0 || j == i)
                    C[i][j] = 1;
 
                // Calculate value using
                // previously stored values
                else
                    C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
            }
        }
 
        return C[n][k];
    }
 
    // A utility function to return
    // minimum of two integers
    static int min(int a, int b) { return (a < b) ? a : b; }
 
    /* Driver program to test above function*/
    public static void main(String args[])
    {
        int n = 5, k = 2;
        System.out.println("Value of C(" + n + "," + k
                           + ") is " + binomialCoeff(n, k));
    }
}
/*This code is contributed by Rajat Mishra*/

Python

# A Dynamic Programming based Python
# Program that uses table C[][]
# to calculate the Binomial Coefficient
 
# Returns value of Binomial Coefficient C(n, k)
 
 
def binomialCoef(n, k):
    C = [[0 for x in range(k+1)] for x in range(n+1)]
 
    # Calculate value of Binomial
    # Coefficient in bottom up manner
    for i in range(n+1):
        for j in range(min(i, k)+1):
            # Base Cases
            if j == 0 or j == i:
                C[i][j] = 1
 
            # Calculate value using
            # previously stored values
            else:
                C[i][j] = C[i-1][j-1] + C[i-1][j]
 
    return C[n][k]
 
 
# Driver program to test above function
n = 5
k = 2
print("Value of C[" + str(n) + "][" + str(k) + "] is "
      + str(binomialCoef(n, k)))
 
# This code is contributed by Bhavya Jain

C＃

// A Dynamic Programming based solution that
// uses table C[][] to calculate the Binomial
// Coefficient
using System;
 
class GFG {
 
    // Returns value of Binomial Coefficient
    // C(n, k)
    static int binomialCoeff(int n, int k)
    {
        int[, ] C = new int[n + 1, k + 1];
        int i, j;
 
        // Calculate value of Binomial
        // Coefficient in bottom up manner
        for (i = 0; i <= n; i++) {
            for (j = 0; j <= Math.Min(i, k); j++) {
                // Base Cases
                if (j == 0 || j == i)
                    C[i, j] = 1;
 
                // Calculate value using previously
                // stored values
                else
                    C[i, j] = C[i - 1, j - 1] + C[i - 1, j];
            }
        }
 
        return C[n, k];
    }
 
    // A utility function to return minimum
    // of two integers
    static int min(int a, int b) { return (a < b) ? a : b; }
 
    /* Driver program to test above function*/
    public static void Main()
    {
        int n = 5, k = 2;
        Console.WriteLine("Value of C(" + n + "," + k
                          + ") is " + binomialCoeff(n, k));
    }
}
 
// This code is contributed by anuj_67.

Javascript

 
// A Dynamic Programming based
// solution that uses table C to
// calculate the Binomial Coefficient
 
// Returns value of Binomial
// Coefficient C(n, k)
function binomialCoeff(n, k)
{
    var C = Array(n + 1).fill(0).map(
      x => Array(k + 1).fill(0));;
    var i, j;
 
    // Calculate  value of Binomial
    // Coefficient in bottom up manner
    for(i = 0; i <= n; i++)
    {
        for(j = 0; j <= min(i, k); j++)
        {
             
            // Base Cases
            if (j == 0 || j == i)
                C[i][j] = 1;
 
            // Calculate value using
            // previously stored values
            else
                C[i][j] = C[i - 1][j - 1] +
                          C[i - 1][j];
        }
    }
    return C[n][k];
}
 
// A utility function to return
// minimum of two integers
function min(a, b)
{
    return (a < b) ? a : b;
}
 
// Driver code
var n = 5, k = 2;
document.write("Value of C(" + n + "," + k +
                     ") is " + binomialCoeff(n, k));
 
// This code is contributed by 29AjayKumar
 

输出

Value of C[5][2] is 10

时间复杂度： O(n*k)
辅助空间： O(n*k)

以下是上述代码的空间优化版本。以下代码仅使用 O(k)。感谢AK提出这个方法。

C++

// C++ program for space optimized Dynamic Programming
// Solution of Binomial Coefficient
#include 
using namespace std;
 
int binomialCoeff(int n, int k)
{
    int C[k + 1];
    memset(C, 0, sizeof(C));
 
    C[0] = 1; // nC0 is 1
 
    for (int i = 1; i <= n; i++) {
        // Compute next row of pascal triangle using
        // the previous row
        for (int j = min(i, k); j > 0; j--)
            C[j] = C[j] + C[j - 1];
    }
    return C[k];
}
 
/* Driver code*/
int main()
{
    int n = 5, k = 2;
    printf("Value of C(%d, %d) is %d ", n, k,
           binomialCoeff(n, k));
    return 0;
}

Java

// JAVA Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
import java.util.*;
 
class GFG {
 
    static int binomialCoeff(int n, int k)
    {
        int C[] = new int[k + 1];
 
        // nC0 is 1
        C[0] = 1;
 
        for (int i = 1; i <= n; i++) {
            // Compute next row of pascal
            // triangle using the previous row
            for (int j = Math.min(i, k); j > 0; j--)
                C[j] = C[j] + C[j - 1];
        }
        return C[k];
    }
 
    /* Driver code  */
    public static void main(String[] args)
    {
        int n = 5, k = 2;
        System.out.printf("Value of C(%d, %d) is %d ", n, k,
                          binomialCoeff(n, k));
    }
}

Python

# Python program for Optimized
# Dynamic Programming solution to
# Binomail Coefficient. This one
# uses the concept of pascal
# Triangle and less memory
 
 
def binomialCoeff(n, k):
 
    # Declaring an empty array
    C = [0 for i in xrange(k+1)]
    C[0] = 1  # since nC0 is 1
 
    for i in range(1, n+1):
 
        # Compute next row of pascal triangle using
        # the previous row
        j = min(i, k)
        while (j > 0):
            C[j] = C[j] + C[j-1]
            j -= 1
 
    return C[k]
 
 
# Driver Code
n = 5
k = 2
print "Value of C(%d,%d) is %d" % (n, k, binomialCoeff(n, k))
 
# This code is contribtued by Nikhil Kumar Singh(nickzuck_007)

C＃

// C# Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
using System;
 
class GFG {
 
    static int binomialCoeff(int n, int k)
    {
        int[] C = new int[k + 1];
 
        // nC0 is 1
        C[0] = 1;
 
        for (int i = 1; i <= n; i++) {
            // Compute next row of pascal
            // triangle using the previous
            // row
            for (int j = Math.Min(i, k); j > 0; j--)
                C[j] = C[j] + C[j - 1];
        }
        return C[k];
    }
 
    /* Driver Code */
    public static void Main()
    {
        int n = 5, k = 2;
        Console.WriteLine("Value of C(" + n + " " + k
                          + ") is " + binomialCoeff(n, k));
    }
}
 
// This code is contribtued by anuj_67.

PHP

 0; $j--)
            $C[$j] = $C[$j] + $C[$j - 1];
    }
    return $C[$k];
}
 
// Driver Code
$n = 5; $k = 2;
echo "Value of C[$n, $k] is ".
        binomialCoeff($n, $k);
     
// This code is contributed by mits.
?>

Javascript

 
// Javascript program for space optimized
// Dynamic Programming
// Solution of Binomial Coefficient
function binomialCoeff(n, k)
{
    let C = new Array(k + 1);
    C.fill(0);
 
    // nC0 is 1
    C[0] = 1;
 
    for(let i = 1; i <= n; i++)
    {
         
        // Compute next row of pascal
        // triangle using the previous
        // row
        for(let j = Math.min(i, k); j > 0; j--)
            C[j] = C[j] + C[j - 1];
    }
    return C[k];
}
 
// Driver code
let n = 5, k = 2;
document.write("Value of C(" + n + " " +
               k + ") is " + binomialCoeff(n, k));
                
// This code is contributed by divyesh072019
 

输出

Value of C(5, 2) is 10 

时间复杂度： O(n*k)
辅助空间： O(k)

解释：
1==========>> n = 0, C(0,0) = 1
1–1========>> n = 1, C(1,0) = 1, C(1,1) = 1
1–2–1======>> n = 2, C(2,0) = 1, C(2,1) = 2, C(2,2) = 1
1–3–3–1====>> n = 3, C(3,0) = 1, C(3,1) = 3, C(3,2) = 3, C(3,3) =1
1–4–6–4–1==>> n = 4, C(4,0) = 1, C(4,1) = 4, C(4,2) = 6, C(4,3) =4，C(4,4)=1
所以这里我的每个循环，使用第 (i-1) 行构建第 i 行的帕斯卡三角形
在任何时候，数组 C 的每个元素都会有一些值(零或更多)，并且在下一次迭代中，这些元素的值来自前一次迭代。
在声明中，
C[j] = C[j] + C[j-1]
右侧代表来自前一次迭代的值(帕斯卡三角形的一行取决于前一行)。左侧表示将通过该语句获得的当前迭代的值。

Let's say we want to calculate C(4, 3), 
i.e. n=4, k=3:

All elements of array C of size 4 (k+1) are
initialized to ZERO.

i.e. C[0] = C[1] = C[2] = C[3] = C[4] = 0;
Then C[0] is set to 1

For i = 1:
C[1] = C[1] + C[0] = 0 + 1 = 1 ==>> C(1,1) = 1

For i = 2:
C[2] = C[2] + C[1] = 0 + 1 = 1 ==>> C(2,2) = 1
C[1] = C[1] + C[0] = 1 + 1 = 2 ==>> C(2,1) = 2

For i=3:
C[3] = C[3] + C[2] = 0 + 1 = 1 ==>> C(3,3) = 1
C[2] = C[2] + C[1] = 1 + 2 = 3 ==>> C(3,2) = 3
C[1] = C[1] + C[0] = 2 + 1 = 3 ==>> C(3,1) = 3

For i=4:
C[4] = C[4] + C[3] = 0 + 1 = 1 ==>> C(4,4) = 1
C[3] = C[3] + C[2] = 1 + 3 = 4 ==>> C(4,3) = 4
C[2] = C[2] + C[1] = 3 + 3 = 6 ==>> C(4,2) = 6
C[1] = C[1] + C[0] = 3 + 1 = 4 ==>> C(4,1) = 4

C(4,3) = 4 is would be the answer in our example.

记忆方法：这个想法是创建一个查找表并遵循递归自顶向下的方法。在计算任何值之前，我们检查它是否已经在查找表中。如果是，我们返回值。否则，我们计算该值并将其存储在查找表中。以下是动态规划的自顶向下方法来寻找二项式系数的值。

C++

// A Dynamic Programming based
// solution that uses
// table dp[][] to calculate
// the Binomial Coefficient
// A naive recursive approach
// with table C++ implementation
#include 
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeffUtil(int n, int k, int** dp)
{
    // If value in lookup table then return
    if (dp[n][k] != -1) //    
        return dp[n][k];
 
    // store value in a table before return
    if (k == 0) {
        dp[n][k] = 1;
        return dp[n][k];
    }
     
    // store value in table before return
    if (k == n) {
        dp[n][k] = 1;
        return dp[n][k];
    }
     
    // save value in lookup table before return
    dp[n][k] = binomialCoeffUtil(n - 1, k - 1, dp) +
               binomialCoeffUtil(n - 1, k, dp);
    return dp[n][k];
}
 
int binomialCoeff(int n, int k)
{
    int** dp; // make a temporary lookup table
    dp = new int*[n + 1];
 
    // loop to create table dynamically
    for (int i = 0; i < (n + 1); i++) {
        dp[i] = new int[k + 1];
    }
 
    // nested loop to initialise the table with -1
    for (int i = 0; i < (n + 1); i++) {
        for (int j = 0; j < (k + 1); j++) {
            dp[i][j] = -1;
        }
    }
 
    return binomialCoeffUtil(n, k, dp);
}
 
/* Driver code*/
int main()
{
    int n = 5, k = 2;
    cout << "Value of C(" << n << ", " << k << ") is "
         << binomialCoeff(n, k) << endl;
    return 0;
}
 
// This is code is contributed by MOHAMMAD MUDASSIR

Java

// A Dynamic Programming based
// solution that uses
// table dp[][] to calculate
// the Binomial Coefficient
// A naive recursive approach
// with table Java implementation
import java.util.*;
class GFG{
 
// Returns value of Binomial
// Coefficient C(n, k)
static int binomialCoeffUtil(int n, int k,
                             Vector []dp)
{
  // If value in lookup table
  // then return
  if (dp[n].get(k) != -1)    
    return dp[n].get(k);
 
  // store value in a table
  // before return
  if (k == 0)
  {
    dp[n].add(k, 1);
    return dp[n].get(k);
  }
 
  // store value in table
  // before return
  if (k == n)
  {
    dp[n].add(k, 1);
    return dp[n].get(k);
  }
 
  // save value in lookup table
  // before return
  dp[n].add(k, binomialCoeffUtil(n - 1,
                                 k - 1, dp) +
               binomialCoeffUtil(n - 1,
                                 k, dp));
  return dp[n].get(k);
}
 
static int binomialCoeff(int n, int k)
{
  // Make a temporary lookup table
  Vector []dp = new Vector[n+1];
 
  // Loop to create table dynamically
  for (int i = 0; i < (n + 1); i++)
  {
    dp[i] = new Vector();
    for(int j = 0; j <= k; j++)
      dp[i].add(-1);
  }
  return binomialCoeffUtil(n, k, dp);
}
 
// Driver code
public static void main(String[] args)
{
  int n = 5, k = 2;
  System.out.print("Value of C(" + n +
                   ", " + k + ") is " +
                   binomialCoeff(n, k) + "\n");
}
}
 
// This code is contributed by Rajput-Ji

蟒蛇3

# A Dynamic Programming based solution
# that uses table dp[][] to calculate
# the Binomial Coefficient. A naive
# recursive approach with table
# Python3 implementation
 
# Returns value of Binomial
# Coefficient C(n, k)
def binomialCoeffUtil(n, k, dp):
     
    # If value in lookup table then return
    if dp[n][k] != -1:
        return dp[n][k]
 
    # Store value in a table before return
    if k == 0:
        dp[n][k] = 1
        return dp[n][k]
     
    # Store value in table before return
    if k == n:
        dp[n][k] = 1
        return dp[n][k]
     
    # Save value in lookup table before return
    dp[n][k] = (binomialCoeffUtil(n - 1, k - 1, dp) +
                binomialCoeffUtil(n - 1, k, dp))
                 
    return dp[n][k]
 
def binomialCoeff(n, k):
     
    # Make a temporary lookup table
    dp = [ [ -1 for y in range(k + 1) ]
                for x in range(n + 1) ]
 
    return binomialCoeffUtil(n, k, dp)
 
# Driver code
n = 5
k = 2
 
print("Value of C(" + str(n) +
               ", " + str(k) + ") is",
               binomialCoeff(n, k))
 
# This is code is contributed by Prateek Gupta

C＃

// C# program for the
// above approach
 
// A Dynamic Programming based
// solution that uses
// table [,]dp to calculate
// the Binomial Coefficient
// A naive recursive approach
// with table C# implementation
using System;
using System.Collections.Generic;
class GFG{
 
// Returns value of Binomial
// Coefficient C(n, k)
static int binomialCoeffUtil(int n, int k,
                             List []dp)
{
  // If value in lookup table
  // then return
  if (dp[n][k] != -1)    
    return dp[n][k];
 
  // store value in a table
  // before return
  if (k == 0)
  {
    dp[n][k] = 1;
    return dp[n][k];
  }
 
  // store value in table
  // before return
  if (k == n)
  {
    dp[n][k] = 1;
    return dp[n][k];
  }
 
  // save value in lookup table
  // before return
  dp[n][k] = binomialCoeffUtil(n - 1,
                               k - 1, dp) +
             binomialCoeffUtil(n - 1,
                               k, dp);
  return dp[n][k];
}
 
static int binomialCoeff(int n, int k)
{
  // Make a temporary lookup table
  List []dp = new List[n + 1];
 
  // Loop to create table dynamically
  for (int i = 0; i < (n + 1); i++)
  {
    dp[i] = new List();
 
    for(int j = 0; j <= k; j++)
      dp[i].Add(-1);
  }
  return binomialCoeffUtil(n, k, dp);
}
 
// Driver code
public static void Main(String[] args)
{
  int n = 5, k = 2;
  Console.Write("Value of C(" + n +
                ", " + k + ") is " +
                binomialCoeff(n, k) + "\n");
}
}
 
// This code is contributed by 29AjayKumar

Javascript

 
// A Dynamic Programming based solution that
// uses table dp[][] to calculate the
// Binomial Coefficient. A naive recursive
// approach with table Javascript implementation
 
// Returns value of Binomial
// Coefficient C(n, k)
function binomialCoeffUtil(n, k, dp)
{
     
    // If value in lookup table
    // then return
    if (dp[n][k] != -1)   
        return dp[n][k];
     
    // Store value in a table
    // before return
    if (k == 0)
    {
        dp[n][k] = 1;
        return dp[n][k];
    }
     
    // Store value in table
    // before return
    if (k == n)
    {
        dp[n][k] = 1;
        return dp[n][k];
    }
     
    // Save value in lookup table
    // before return
    dp[n][k] = binomialCoeffUtil(n - 1,
                                 k - 1, dp) +
               binomialCoeffUtil(n - 1,
                                 k, dp);
    return dp[n][k];
}
 
function binomialCoeff(n, k)
{
     
    // Make a temporary lookup table
    let dp = new Array(n + 1);
     
    // Loop to create table dynamically
    for(let i = 0; i < (n + 1); i++)
    {
        dp[i] = [];
        for(let j = 0; j <= k; j++)
            dp[i].push(-1);
    }
    return binomialCoeffUtil(n, k, dp);
}
 
// Driver code
let n = 5, k = 2;
document.write("Value of C(" + n +
               ", " + k + ") is " +
               binomialCoeff(n, k) + "\n");
 
// This code is contributed by avanitrachhadiya2155
 

输出

Value of C(5, 2) is 10

取消分子和分母之间的因数：

nCr = (n-r+1)*(n-r+2)*….*n / (r!)

创建一个从 n-r+1 到 n 的数字数组 arr，其大小为 r。由于 nCr 始终是整数，因此分母中的所有数字都应与分子(由 arr 表示)的乘积相抵消。

对于 i = 1 到 i = r，

搜索 arr，如果 arr[j] 和 i 有 gcd>1，则除以 gcd，当 i 变为 1 时，停止搜索

现在，答案只是 arr 的乘积，其值 mod 10^9+7 可以使用单次传递找到，公式使用 (a*b)%mod = (a%mod * b%mod)%mod

Java

import java.util.*;
class GFG
{
    static int gcd(int a, int b) // function to find gcd of two numbers in O(log(min(a,b)))
    {
        if(b==0) return a;
        return gcd(b,a%b);
    }
    static int nCr(int n, int r)
    {
        if(r>n) // base case
            return 0;
        if(r>n-r) // C(n,r) = C(n,n-r) better time complexity for lesser r value
            r = n-r;
        int mod = 1000000007;
        int[] arr = new int[r]; // array of elements from n-r+1 to n
        for(int i=n-r+1; i<=n; i++)
        {
            arr[i+r-n-1] = i;
        }
        long ans = 1;
        for(int k=1;k1
        {
            int j=0, i=k;
            while(j1)
                {
                    arr[j] /= x; // if gcd>1, divide both by gcd
                    i /= x;
                }
                if(i==1)
                    break; // if i becomes 1, no need to search arr
                j += 1;
            }
        }
        for(int i : arr) // single pass to multiply the numerator
            ans = (ans*i)%mod;
        return (int)ans;
    }
    // Driver code
    public static void main(String[] args)
    {
        int n = 5, r = 2;
        System.out.print("Value of C(" +  n+ ", " +  r+ ") is "
             +nCr(n, r) +"\n");
    }
}
// This code is contributed by Gautam Wadhwani

蟒蛇3

import math
class GFG:
    def nCr(self, n, r):
        def gcd(a,b): # function to find gcd of two numbers in O(log(min(a,b)))
            if b==0: # base case
                return a
            return gcd(b,a%b)
        if r>n:
            return 0
        if r>n-r: # C(n,r) = C(n,n-r) better time complexity for lesser r value
            r = n-r
        mod = 10**9 + 7
        arr = list(range(n-r+1,n+1)) # array of elements from n-r+1 to n
        ans = 1
        for i in range(1,r+1): # for numbers from 1 to r find arr[j] such that gcd(i,arr[j])>1
            j=0
            while j1:
                    arr[j] //= x # if gcd>1, divide both by gcd
                    i //= x
                if arr[j]==1: # if element becomes 1, its of no use anymore so delete from arr
                    del arr[j]
                    j -= 1
                if i==1:
                    break # if i becomes 1, no need to search arr
                j += 1
        for i in arr: # single pass to multiply the numerator
            ans = (ans*i)%mod
        return ans
     # Driver code
n = 5
k = 2
ob = GFG()
print("Value of C(" + str(n) +
               ", " + str(k) + ") is",
               ob.nCr(n, k))
  
# This is code is contributed by Gautam Wadhwani

C＃

using System;
 
class GFG{
 
// Function to find gcd of two numbers
// in O(log(min(a,b)))  
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
         
    return gcd(b, a % b);
}
 
static int nCr(int n, int r)
{
     
    // Base case
    if (r > n)
        return 0;
         
    // C(n,r) = C(n,n-r) better time
    // complexity for lesser r value
    if (r > n - r)
        r = n - r;
         
    int mod = 1000000007;
     
    // Array of elements from n-r+1 to n
    int[] arr = new int[r];
    for(int i = n - r + 1; i <= n; i++)
    {
        arr[i + r - n - 1] = i;
    }
    long ans = 1;
     
    // For numbers from 1 to r find arr[j]
    // such that gcd(i,arr[j])>1
    for(int k = 1; k < r + 1; k++)
    {
        int j = 0, i = k;
        while (j < arr.Length)
        {
            int x = gcd(i,arr[j]);
            if (x > 1)
            {
                 
                // If gcd>1, divide both by gcd
                arr[j] /= x;
                i /= x;
            }
             
            if (i == 1)
             
                // If i becomes 1, no need
                // to search arr
                break;
                 
            j += 1;
        }
    }
     
    // Single pass to multiply the numerator
    foreach(int i in arr)
        ans = (ans * i) % mod;
         
    return (int)ans;
}
 
// Driver code
static public void Main()
{
    int n = 5, r = 2;
    Console.WriteLine("Value of C(" +  n +
                      ", " +  r + ") is " +
                      nCr(n, r) + "\n");
}
}
 
// This code is contributed by rag2127

输出

Value of C(5, 2) is 10

时间复杂度： O(( min(r, nr)^2 ) * log(n))，当 n >> r 或 n >> (nr) 时有用

辅助空间： O(min(r, nr))

以对数时间查看 GCD

使用 Eratosthenes 筛法对从 1 到 n 的每个数字进行质因数分解：

1. 创建一个大小为 n+1 的数组 SPF 到从 1 到 n 的每个数字的最小素因数

Set SPF[i] = i for all i = 1 to i = n

2. 使用埃拉托色尼筛子：

for i = 2 to i = n:
    if i is prime,
       for all multiples j of i, j<=n:
           if SPF[j] equals j, set SPF[j] = i

3. 一旦我们知道从 1 到 n 的每个数的 SPF，我们可以在 O(log(n)) 时间内使用 SPF 的递归除法找到从 1 到 n 的任何数的素因数分解，直到该数变为 1

Now, nCr  =  (n-r+1)*(r+2)* ... *(n) / (r)!

4.创建一个字典(或hashmap)来存储nCr实际值的素数分解中每个素数的频率。

5. 因此，只需计算 nCr 中每个素数的频率并将它们相乘，以提高其频率的幂。

6. 对于分子，迭代 i = n-r+1 到 i = n，对于 i 的所有质因子，将它们的频率存储在字典中。

( prime_pow[prime_factor] += freq_in_i )

7. 对于分母，迭代 i = 1 到 i = r，现在减去频率而不是增加频率。

8. 现在，遍历字典并将答案乘以 (prime ^ prime_pow[prime]) % (10^9 + 7)

ans = (ans * pow(prime, prime_pow[prime], mod) ) % mod

Java

import java.util.*;
class GFG {
   
    // pow(base,exp,mod) is used to find
    // (base^exp)%mod fast -> O(log(exp))
    static long pow(long b, long exp, long mod)
    {
        long ret = 1;
        while (exp > 0) {
            if ((exp & 1) > 0)
                ret = (ret * b) % mod;
            b = (b * b) % mod;
            exp >>= 1;
        }
        return ret;
    }
    static int nCr(int n, int r)
    {
        if (r > n) // base case
            return 0;
       
        // C(n,r) = C(n,n-r) Complexity for
        // this code is lesser for lower n-r
        if (n - r > r)
            r = n - r;
       
        // list to smallest prime factor
        // of each number from 1 to n
        int[] SPF = new int[n + 1];
       
        // set smallest prime factor of each
        // number as itself
        for (int i = 1; i <= n; i++)
            SPF[i] = i;
       
        // set smallest prime factor of all
        // even numbers as 2
        for (int i = 4; i <= n; i += 2)
            SPF[i] = 2;
       
        for (int i = 3; i * i < n + 1; i += 2)
        {
             
            // Check if i is prime
            if (SPF[i] == i)
            {
               
                // All multiples of i are
                // composite (and divisible by
                // i) so add i to their prime
                // factorization getpow(j,i)
                // times
                for (int j = i * i; j < n + 1; j += i)
                    if (SPF[j] == j) {
                        SPF[j] = i;
                    }
            }
        }
       
       // Hash Map to store power of
       // each prime in C(n,r)
       Map prime_pow
            = new HashMap<>();
       
        // For numerator count frequency of each prime factor
        for (int i = r + 1; i < n + 1; i++)
        {
            int t = i;
             
            // Recursive division to find
            // prime factorization of i
            while (t > 1)
            {
                prime_pow.put(SPF[t],
                    prime_pow.getOrDefault(SPF[t], 0) + 1);
                t /= SPF[t];
            }
        }
       
        // For denominator subtract the power of
        // each prime factor
        for (int i = 1; i < n - r + 1; i++)
        {
            int t = i;
           
            // Recursive division to find
            // prime factorization of i
            while (t > 1)
            {
                prime_pow.put(SPF[t],
                              prime_pow.get(SPF[t]) - 1);
                t /= SPF[t];
            }
        }
       
        // long because mod is large and a%mod
        // * b%mod can overflow int
        long ans = 1, mod = 1000000007;
       
        // use (a*b)%mod = (a%mod * b%mod)%mod
        for (int i : prime_pow.keySet())
           
            // pow(base,exp,mod) is used to
            // find (base^exp)%mod fast
            ans = (ans * pow(i, prime_pow.get(i), mod))
                  % mod;
        return (int)ans;
    }
   
    // Driver code
    public static void main(String[] args)
    {
        int n = 5, r = 2;
        System.out.print("Value of C(" + n + ", " + r
                         + ") is " + nCr(n, r) + "\n");
    }
}
// This code is contributed by Gautam Wadhwani

蟒蛇3

# Python code for the above approach
import math
 
class GFG:
    def nCr(self, n, r):
       
        # Base case
        if r > n: 
            return 0
           
        # C(n,r) = C(n,n-r) Complexity for this
        # code is lesser for lower n-r
        if n - r > r: 
            r = n - r
             
        # List to store smallest prime factor
        # of every number from 1 to n
        SPF = [i for i in range(n+1)]
        for i in range(4, n+1, 2):
           
            # set smallest prime factor of
            # all even numbers as 2
            SPF[i] = 2
     
        for i in range(3, n+1, 2): 
         
            if i*i > n:
                break
               
            # Check if i is prime
            if SPF[i] == i: 
                 
                # All multiples of i are composite
                # (and divisible by i) so add i to
                # their prime factorization getpow(j,i) times
                for j in range(i*i, n+1, i):
                    if SPF[j] == j:
                       
                        # set smallest prime factor
                        # of j to i only if it is
                        # not previously set
                        SPF[j] = i
         
         # dictionary to store power of each prime in C(n,r)
        prime_pow = {} 
         
        # For numerator count frequency
        # of each prime factor
        for i in range(r+1, n+1):
            t = i
             
            # Recursive division to
            # find prime factorization of i
            while t > 1:
                if not SPF[t] in prime_pow:
                    prime_pow[SPF[t]] = 1
                else:
                    prime_pow[SPF[t]] += 1
                t //= SPF[t]
         
        # For denominator subtract the
        # power of each prime factor
        for i in range(1, n-r+1): 
            t = i
             
            # Recursive division to
            # find prime factorization of i
            while t > 1: 
                prime_pow[SPF[t]] -= 1
                t //= SPF[t]
        ans = 1
        mod = 10**9 + 7
         
         # Use (a*b)%mod = (a%mod * b%mod)%mod
        for i in prime_pow:
           
            # pow(base,exp,mod) is used to
            # find (base^exp)%mod fast
            ans = (ans*pow(i, prime_pow[i], mod)) % mod
        return ans
 
 
# Driver code
n = 5
k = 2
ob = GFG()
print("Value of C(" + str(n) +
      ", " + str(k) + ") is",
      ob.nCr(n, k))
 
# This is code is contributed by Gautam Wadhwani

输出

Value of C(5, 2) is 10

时间复杂度： O(n*log(n))，当 r->n/2 时非常有用

辅助空间： O(n)

请参阅 O(log(n)) 中的质数分解

另一种方法：(模块化反演技术)

1. nCr 的通式为 ( n*(n-1)*(n-2)* … *(n-r+1) ) / (r!)。我们可以直接用这个公式来求nCr。但这会溢出。我们需要找到 nCr mod m 以便它不会溢出。我们可以很容易地用模算术的公式来做到这一点。

for the  n*(n-1)*(n-2)* ... *(n-r+1) part we can use the formula,
(a*b) mod m = ((a % m) * (b % m)) % m

2.对于1/r！部分，我们需要找到从 1 到 r 的每个数字的模逆。然后使用上面相同的公式和 1 到 r 的模逆。我们可以使用公式在 O(r) 时间内找到模逆，

inv[1] = 1
inv[i] = − ⌊m/i⌋ * inv[m mod i] mod m
To use this formula, m has to be a prime.

在练习题中，我们需要用模 1000000007 显示答案，这是一个质数。

所以，这种技术会奏效。

C++

// C++ program for the above approach
#include
using namespace std;
 
// Function to find binomial
// coefficient
int binomialCoeff(int n, int r)
{
 
    if (r > n)
        return 0;
    long long int m = 1000000007;
    long long int inv[r + 1] = { 0 };
    inv[0] = 1;
    if(r+1>=2)
    inv[1] = 1;
 
    // Getting the modular inversion
    // for all the numbers
    // from 2 to r with respect to m
    // here m = 1000000007
    for (int i = 2; i <= r; i++) {
        inv[i] = m - (m / i) * inv[m % i] % m;
    }
 
    int ans = 1;
 
    // for 1/(r!) part
    for (int i = 2; i <= r; i++) {
        ans = ((ans % m) * (inv[i] % m)) % m;
    }
 
    // for (n)*(n-1)*(n-2)*...*(n-r+1) part
    for (int i = n; i >= (n - r + 1); i--) {
        ans = ((ans % m) * (i % m)) % m;
    }
    return ans;
}
 
/* Driver code*/
int main()
{
    int n = 5, r = 2;
    cout << "Value of C(" << n << ", " << r << ") is "
         << binomialCoeff(n, r) << endl;
    return 0;
}

Java

// JAVA program for the above approach
import java.util.*;
class GFG
{
 
// Function to find binomial
// coefficient
static int binomialCoeff(int n, int r)
{
 
    if (r > n)
    return 0;
    long  m = 1000000007;
    long  inv[] = new long[r + 1];
    inv[0] = 1;
    if(r+1>=2)
    inv[1] = 1;
 
    // Getting the modular inversion
    // for all the numbers
    // from 2 to r with respect to m
    // here m = 1000000007
    for (int i = 2; i <= r; i++) {
        inv[i] = m - (m / i) * inv[(int) (m % i)] % m;
    }
 
    int ans = 1;
 
    // for 1/(r!) part
    for (int i = 2; i <= r; i++) {
        ans = (int) (((ans % m) * (inv[i] % m)) % m);
    }
 
    // for (n)*(n-1)*(n-2)*...*(n-r+1) part
    for (int i = n; i >= (n - r + 1); i--) {
        ans = (int) (((ans % m) * (i % m)) % m);
    }
    return ans;
}
 
/* Driver code*/
public static void main(String[] args)
{
    int n = 5, r = 2;
    System.out.print("Value of C(" +  n+ ", " +  r+ ") is "
         +binomialCoeff(n, r) +"\n");
}
}
 
// This code contributed by Rajput-Ji

蟒蛇3

# Python3 program for the above approach
 
# Function to find binomial
# coefficient
def binomialCoeff(n, r):
     
    if (r > n):
        return 0
         
    m = 1000000007
    inv = [0 for i in range(r + 1)]
    inv[0] = 1;
    if(r+1>=2)
    inv[1] = 1;
 
    # Getting the modular inversion
    # for all the numbers
    # from 2 to r with respect to m
    # here m = 1000000007
    for i in range(2, r + 1):
        inv[i] = m - (m // i) * inv[m % i] % m
 
    ans = 1
 
    # for 1/(r!) part
    for i in range(2, r + 1):
        ans = ((ans % m) * (inv[i] % m)) % m
 
    # for (n)*(n-1)*(n-2)*...*(n-r+1) part
    for i in range(n, n - r, -1):
        ans = ((ans % m) * (i % m)) % m
         
    return ans
 
# Driver code
n = 5
r = 2
 
print("Value of C(" ,n , ", " , r ,
      ") is ",binomialCoeff(n, r))
 
# This code is contributed by rohan07

C＃

// C# program for the above approach
using System;
 
public class GFG
{
 
// Function to find binomial
// coefficient
static int binomialCoeff(int n, int r)
{
 
    if (r > n)
    return 0;
    long  m = 1000000007;
    long  []inv = new long[r + 1];
    inv[0] = 1;
    if(r+1>=2)
    inv[1] = 1;
 
    // Getting the modular inversion
    // for all the numbers
    // from 2 to r with respect to m
    // here m = 1000000007
    for (int i = 2; i <= r; i++) {
        inv[i] = m - (m / i) * inv[(int) (m % i)] % m;
    }
 
    int ans = 1;
 
    // for 1/(r!) part
    for (int i = 2; i <= r; i++) {
        ans = (int) (((ans % m) * (inv[i] % m)) % m);
    }
 
    // for (n)*(n-1)*(n-2)*...*(n-r+1) part
    for (int i = n; i >= (n - r + 1); i--) {
        ans = (int) (((ans % m) * (i % m)) % m);
    }
    return ans;
}
 
/* Driver code*/
public static void Main(String[] args)
{
    int n = 5, r = 2;
    Console.Write("Value of C(" +  n+ ", " +  r+ ") is "
         +binomialCoeff(n, r) +"\n");
}
}
 
 
 
// This code is contributed by 29AjayKumar

Javascript

        // JavaScript Program for the above approach
 
        // Function to find binomial
        // coefficient
        function binomialCoeff(n, r) {
 
            if (r > n)
                return 0;
            let m = 1000000007;
            let inv = new Array(r + 1).fill(0);
            inv[0] = 1;
            if (r + 1 >= 2)
                inv[1] = 1;
 
            // Getting the modular inversion
            // for all the numbers
            // from 2 to r with respect to m
            // here m = 1000000007
            for (let i = 2; i <= r; i++) {
                inv[i] = m - Math.floor(m / i) * inv[m % i] % m;
            }
 
            let ans = 1;
 
            // for 1/(r!) part
            for (let i = 2; i <= r; i++) {
                ans = ((ans % m) * (inv[i] % m)) % m;
            }
 
            // for (n)*(n-1)*(n-2)*...*(n-r+1) part
            for (let i = n; i >= (n - r + 1); i--) {
                ans = ((ans % m) * (i % m)) % m;
            }
            return ans;
        }
 
        /* Driver code*/
        let n = 5, r = 2;
        document.write("Value of C(" + n + ", " + r + ") is "
            + binomialCoeff(n, r) + "<br>");
 
    // This code is contributed by Potta Lokesh
    

输出

Value of C(5, 2) is 10

时间复杂度： O(n+k)

辅助空间： O(k)

有关空间和时间效率的二项式系数，请参阅此内容

参考：
http://www.csl.mtu.edu/cs4321/www/Lectures/Lecture%2015%20-%20Dynamic%20Programming%20Binomial%20Coefficients.htm

https://cp-algorithms.com/algebra/module-inverse.html

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