最大二项式系数项值

给定正整数n 。任务是在所有二项式系数中找到最大系数项。
二项式系数序列为
ⁿ C ₀ ， ⁿ C ₁ ， ⁿ C ₂ ，…。， ⁿ C _r ，…。， ⁿ C _n-2 ， ⁿ C _n-1 ， ⁿ C _n
任务是找到ⁿ C _{r的最大值}。
例子：

Input : n = 4
Output : 6
4C0 = 1
4C1 = 4
4C2 = 6
4C3 = 1
4C4 = 1
So, maximum coefficient value is 6.

Input : n = 3
Output : 3

方法1 ：(强力)
想法是找到二项式系数系列的所有值并找到该系列的最大值。
以下是此方法的实现：

C++

// CPP Program to find maximum binomial coefficient
// term
#include
using namespace std;
 
// Return maximum binomial coefficient term value.
int maxcoefficientvalue(int n)
{
    int C[n+1][n+1];
  
    // Calculate value of Binomial Coefficient in
    // bottom up manner
    for (int i = 0; i <= n; i++)
    {
        for (int j = 0; j <= min(i, n); 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];
        }
    }
         
    // finding the maximum value.
    int maxvalue = 0;
    for (int i = 0; i <= n; i++)
        maxvalue = max(maxvalue, C[n][i]);
         
    return maxvalue;
}
 
// Driven Program
int main()
{
    int n = 4;
    cout << maxcoefficientvalue(n) << endl;
    return 0;
}

Java

// Java Program to find
// maximum binomial
// coefficient term
import java.io.*;
 
class GFG
{
// Return maximum binomial
// coefficient term value.
static int maxcoefficientvalue(int n)
{
    int [][]C = new int[n + 1][n + 1];
 
    // Calculate value of
    // Binomial Coefficient 
    // in bottom up manner
    for (int i = 0; i <= n; i++)
    {
        for (int j = 0;
                 j <= Math.min(i, n); 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];
        }
    }
         
    // finding the
    // maximum value.
    int maxvalue = 0;
     
    for (int i = 0; i <= n; i++)
        maxvalue = Math.max(maxvalue, C[n][i]);
         
    return maxvalue;
}
 
// Driver Code
public static void main (String[] args)
{
    int n = 4;
    System.out.println(maxcoefficientvalue(n));
}
}
 
// This code is contributed by ajit

Python3

# Python3 Program to find
# maximum binomial
# coefficient term
 
# Return maximum binomial
# coefficient term value.
def maxcoefficientvalue(n):
    C = [[0 for x in range(n + 1)]
            for y 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, n) + 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]);
     
    # finding the maximum value.
    maxvalue = 0;
    for i in range(n + 1):
        maxvalue = max(maxvalue, C[n][i]);
     
    return maxvalue;
 
# Driver Code
n = 4;
print(maxcoefficientvalue(n));
 
# This code is contributed by mits

C#

// C# Program to find maximum binomial coefficient
// term
using System;
 
public class GFG {
         
    // Return maximum binomial coefficient term value.
    static int maxcoefficientvalue(int n)
    {
        int [,]C = new int[n+1,n+1];
     
        // Calculate value of Binomial Coefficient in
        // bottom up manner
        for (int i = 0; i <= n; i++)
        {
            for (int j = 0; j <= Math.Min(i, n); 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];
            }
        }
             
        // finding the maximum value.
        int maxvalue = 0;
         
        for (int i = 0; i <= n; i++)
            maxvalue = Math.Max(maxvalue, C[n,i]);
             
        return maxvalue;
    }
     
    // Driven Program
 
    static public void Main ()
    {
         
        int n = 4;
         
        Console.WriteLine(maxcoefficientvalue(n));
    }
}
 
// This code is contributed by vt_m.

Javascript

 
// JavaScript Program to find
// maximum binomial
// coefficient term
               
    // Returns value of
    // Binomial Coefficient
    // C(n, k)
    function binomialCoeff(n, k)
    {
        let C = new Array(n+1);
         
        // Loop to create 2D array using 1D array
    for (let i = 0; i < C.length; i++) {
        C[i] = new Array(2);
    }
       
        // Calculate value of
        // Binomial Coefficient
        // in bottom up manner
        for (let i = 0; i <= n; i++)
        {
            for (let 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];
    }
       
    // Return maximum
    // binomial coefficient
    // term value.
    function maxcoefficientvalue(n)
    {
           
        // if n is even
        if (n % 2 == 0)
            return binomialCoeff(n, n / 2);
               
        // if n is odd
        else
            return binomialCoeff(n, (n + 1) / 2);
    }
 
// Driver Code
        let n = 4;  
        document.write(maxcoefficientvalue(n));
   
  // This code is contributed by avijitmondal1998..

输出：

6

方法2 ：(使用公式)

证明，

Expansion of (x + y)ⁿ are: 
ⁿC₀ xⁿ y⁰, ⁿC₁ x^n-1 y¹, ⁿC₂ x^n-2 y², …., ⁿC_r x^n-r y^r, …., ⁿC_n-2 x² y^n-2, ⁿC_n-1 x¹ y^n-1, ⁿC_n x⁰ yⁿ
So, putting x = 1 and y = 1, we get binomial coefficient, 
ⁿC₀, ⁿC₁, ⁿC₂, …., ⁿC_r, …., ⁿC_n-2, ⁿC_n-1, ⁿC_n
Let term t_i+1 contains the greatest value in (x + y)ⁿ. Therefore, 
t_r+1 >= t_r 
ⁿC_r x^n-r y^r >= ⁿC_r-1 x^n-r+1 y^r-1
Putting x = 1 and y = 1, 
ⁿC_r >= ⁿC_r-1 
ⁿC_r/ⁿC_r-1 >= 1 
(using ⁿC_r/ⁿC_r-1 = (n-r+1)/r) 
(n-r+1)/r >= 1 
(n+1)/r – 1 >= 1 
(n+1)/r >= 2 
(n+1)/2 >= r
Therefore, r should be less than equal to (n+1)/2. 
And r should be integer. So, we get maximum coefficient for r equals to: 
(1) n/2, when n is even. 
(2) (n+1)/2 or (n-1)/2, when n is odd. 
 

C++

// CPP Program to find maximum binomial coefficient term
#include
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff(int n, int k)
{
    int C[n+1][k+1];
  
    // Calculate value of Binomial Coefficient
    // in bottom up manner
    for (int i = 0; i <= n; i++)
    {
        for (int 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];
}
 
// Return maximum binomial coefficient term value.
int maxcoefficientvalue(int n)
{
    // if n is even
    if (n%2 == 0)
        return binomialCoeff(n, n/2);
         
    // if n is odd
    else
        return binomialCoeff(n, (n+1)/2);
}
 
// Driven Program
int main()
{
    int n = 4;
    cout << maxcoefficientvalue(n) << endl;
    return 0;
}

Java

// Java Program to find
// maximum binomial
// coefficient term
import java.io.*;
 
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];
     
        // Calculate value of
        // Binomial Coefficient
        // in bottom up manner
        for (int i = 0; i <= n; i++)
        {
            for (int 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];
    }
     
    // Return maximum
    // binomial coefficient
    // term value.
    static int maxcoefficientvalue(int n)
    {
         
        // if n is even
        if (n % 2 == 0)
            return binomialCoeff(n, n / 2);
             
        // if n is odd
        else
            return binomialCoeff(n, (n + 1) / 2);
    }
     
    // Driver Code
    public static void main(String[] args)
    {
        int n = 4;
     
        System.out.println(maxcoefficientvalue(n));
    }
}
 
// This code is contributed
// by akt_mit

Python3

# Python3 Program to find
# maximum binomial
# coefficient term
# Returns value of
# Binomial Coefficient C(n, k)
def binomialCoeff(n, k):
 
    C=[[0 for x in range(k+1)] for y 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];
 
 
# Return maximum binomial
# coefficient term value.
def maxcoefficientvalue(n):
    # if n is even
    if (n % 2 == 0):
        return binomialCoeff(n, int(n / 2));
         
    # if n is odd
    else:
        return binomialCoeff(n, int((n + 1) / 2));
 
# Driver Code
if __name__=='__main__':
    n = 4;
    print(maxcoefficientvalue(n));
 
# This code is contributed by mits

C＃

// C# Program to find maximum binomial
// coefficient term
using System;
 
public 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];
     
        // Calculate value of Binomial
        // Coefficient in bottom up manner
        for (int i = 0; i <= n; i++)
        {
            for (int 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];
    }
     
    // Return maximum binomial coefficient
    // term value.
    static int maxcoefficientvalue(int n)
    {
         
        // if n is even
        if (n % 2 == 0)
            return binomialCoeff(n, n/2);
             
        // if n is odd
        else
            return binomialCoeff(n, (n + 1) / 2);
    }
     
    // Driven Program
    static public void Main ()
    {
         
        int n = 4;
         
        Console.WriteLine(maxcoefficientvalue(n));
    }
}
 
// This code is contributed by vt_m.

输出：

6