长度为N的二进制字符串的计数，以使1的频率超过0的频率

给定整数N ，任务是找到长度为N的二进制字符串数，以使1的频率大于0的频率。

例子：

Input: N = 2
Output: 1
Explanation: Count of binary strings of length 2 is 4 i.e. {“00”, “01”, “10”, “11”}. 
The only string having frequency of 1’s greater than that of 0’s is “11”.

Input: N = 3
Output: 4
Explanation: Count of binary strings of length 3 is 8 i.e. {“000”, “001”, “010”, “011”, “100”, “101”, “110”, “111”}.
Among them, the strings having frequency of 1’s greater than 0’s are {“011”, “101”, “110”, “111”}.

天真的方法：解决此问题的最简单方法是生成所有长度为N的二进制字符串，并对每个字符串进行迭代以找到1和0的频率。如果1的频率大于0的频率，则增加计数器。最后，打印计数。

时间复杂度： O(N * 2 ^N )
辅助空间： O(1)

高效方法：要观察上述方法，需要进行以下观察：

S_Total = Total Number of binary strings of length N = 2^N 
 

S_equal = Number of binary string of length N having same frequency of 0’s and 1’s. 
S₁ = Number of binary strings of length N having frequency of 1’s greater than 0’s. 
S₀ = Number of binary strings of length N having frequency of 0’s greater than 1’s. 
S_total = S_equal + S₁ + S₀ 
 

1. 对于S _1中的每个字符串， S _0中都存在一个字符串_。
   假设“ 1110 ”是S _1中的字符串，则其在S _0中的对应字符串将是“ 0001 ”。同样，对于S _1中的每个字符串 在S _0中存在一个字符串。因此， S ₁ = S ₀ (每N个)。
2. 如果N为奇数，则S_等于0 。
3. 如果N是偶数，则S_等于C(N，N / 2) 。

下面是上述方法的实现：

C++

// C++ Program to implement
// the above approach
#include 
using namespace std;
 
// Function to calculate
// and return the value of
// Binomial Coefficient C(n, k)
unsigned long int binomialCoeff(unsigned long int n,
                                unsigned long int k)
{
 
    unsigned long int res = 1;
 
    // Since C(n, k) = C(n, n-k)
    if (k > n - k)
        k = n - k;
 
    // Calculate the value of
    // [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
    for (int i = 0; i < k; ++i) {
        res *= (n - i);
        res /= (i + 1);
    }
 
    return res;
}
 
// Function to return the count of
// binary strings of length N such
// that frequency of 1's exceed that of 0's
unsigned long int countOfString(int N)
{
    // Count of N-length binary strings
    unsigned long int Stotal = pow(2, N);
 
    // Count of N- length binary strings
    // having equal count of 0's and 1's
    unsigned long int Sequal = 0;
 
    // For even length strings
    if (N % 2 == 0)
        Sequal = binomialCoeff(N, N / 2);
 
    unsigned long int S1 = (Stotal - Sequal) / 2;
    return S1;
}
 
// Driver Code
int main()
{
    int N = 3;
    cout << countOfString(N);
    return 0;
}

Java

// Java program to implement
// the above approach
import java.util.*;
 
class GFG{
 
// Function to calculate
// and return the value of
// Binomial Coefficient C(n, k)
static int binomialCoeff(int n, int k)
{
    int res = 1;
 
    // Since C(n, k) = C(n, n-k)
    if (k > n - k)
        k = n - k;
 
    // Calculate the value of
    // [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
    for(int i = 0; i < k; ++i)
    {
        res *= (n - i);
        res /= (i + 1);
    }
    return res;
}
 
// Function to return the count of
// binary Strings of length N such
// that frequency of 1's exceed that of 0's
static int countOfString(int N)
{
     
    // Count of N-length binary Strings
    int Stotal = (int) Math.pow(2, N);
 
    // Count of N- length binary Strings
    // having equal count of 0's and 1's
    int Sequal = 0;
 
    // For even length Strings
    if (N % 2 == 0)
        Sequal = binomialCoeff(N, N / 2);
 
    int S1 = (Stotal - Sequal) / 2;
    return S1;
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 3;
    System.out.print(countOfString(N));
}
}
 
// This code is contributed by Amit Katiyar

Python3

# Python3 program to implement
# the above approach
 
# Function to calculate
# and return the value of
# Binomial Coefficient C(n, k)
def binomialCoeff(n, k):
 
    res = 1
 
    # Since C(n, k) = C(n, n-k)
    if(k > n - k):
        k = n - k
 
    # Calculate the value of
    # [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
    for i in range(k):
        res *= (n - i)
        res //= (i + 1)
 
    return res
 
# Function to return the count of
# binary strings of length N such
# that frequency of 1's exceed that of 0's
def countOfString(N):
 
    # Count of N-length binary strings
    Stotal = pow(2, N)
 
    # Count of N- length binary strings
    # having equal count of 0's and 1's
    Sequal = 0
 
    # For even length strings
    if(N % 2 == 0):
        Sequal = binomialCoeff(N, N // 2)
 
    S1 = (Stotal - Sequal) // 2
 
    return S1
 
# Driver Code
N = 3
 
print(countOfString(N))
 
# This code is contributed by Shivam Singh

C#

// C# program to implement
// the above approach
using System;
class GFG{
 
// Function to calculate
// and return the value of
// Binomial Coefficient C(n, k)
static int binomialCoeff(int n, int k)
{
    int res = 1;
 
    // Since C(n, k) = C(n, n-k)
    if (k > n - k)
        k = n - k;
 
    // Calculate the value of
    // [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
    for(int i = 0; i < k; ++i)
    {
        res *= (n - i);
        res /= (i + 1);
    }
    return res;
}
 
// Function to return the count of
// binary Strings of length N such
// that frequency of 1's exceed that of 0's
static int countOfString(int N)
{
     
    // Count of N-length binary Strings
    int Stotal = (int) Math.Pow(2, N);
 
    // Count of N- length binary Strings
    // having equal count of 0's and 1's
    int Sequal = 0;
 
    // For even length Strings
    if (N % 2 == 0)
        Sequal = binomialCoeff(N, N / 2);
 
    int S1 = (Stotal - Sequal) / 2;
    return S1;
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 3;
    Console.Write(countOfString(N));
}
}
 
// This code is contributed by sapnasingh4991

Javascript

    // Javascript program to implement
    // the above approach
     
    // Function to calculate
    // and return the value of
    // Binomial Coefficient C(n, k)
    function binomialCoeff(n, k)
    {
        let res = 1;
 
        // Since C(n, k) = C(n, n-k)
        if (k > n - k)
            k = n - k;
 
        // Calculate the value of
        // [n*(n-1)*---*(n-k+1)] / [k*(k-1)*---*1]
        for(let i = 0; i < k; ++i)
        {
            res *= (n - i);
            res /= (i + 1);
        }
        return res;
    }
 
    // Function to return the count of
    // binary Strings of length N such
    // that frequency of 1's exceed that of 0's
    function countOfString(N)
    {
 
        // Count of N-length binary Strings
        let Stotal = Math.pow(2, N);
 
        // Count of N- length binary Strings
        // having equal count of 0's and 1's
        let Sequal = 0;
 
        // For even length Strings
        if (N % 2 == 0)
            Sequal = binomialCoeff(N, N / 2);
 
        let S1 = (Stotal - Sequal) / 2;
        return S1;
    }
     
    let N = 3;
    document.write(countOfString(N));
 
    // This code is contributed by divyeshrabadiya07.

输出：

4

时间复杂度： O(N)
辅助空间： O(1)