# Count of ways to choose N people containing at least 4 boys and 1 girl from P boys and Q girls

Given integers **N**, **P**, and **Q** the task is to find the number of ways to form a group of **N people** having **at least 4 boys **and** 1 girls **from **P boys** and **Q girls**.

**Examples:**

Input:P = 5, Q = 2, N = 5Output:10Explanation:Suppose given pool is {m1, m2, m3, m4, m5} and {w1, w2}. Then possible combinations are:m1 m2 m3 m4 w1

m2 m3 m4 m5 w1

m1 m3 m4 m5 w1

m1 m2 m4 m5 w1

m1 m2 m3 m5 w1

m1 m2 m3 m4 w2

m2 m3 m4 m5 w2

m1 m3 m4 m5 w2

m1 m2 m4 m5 w2

m1 m2 m3 m5 w2Hence the count is 10.

Input:P = 5, Q = 2, N = 6Output:7

**Approach:** This problem is based on **combinatorics** where we need to select at least **4** boys out of **1** boys available, and at least **Y** girls out of **Q** girls available, so that total people selected are N.

Consider the example:

P = 5, Q = 2, N = 6

In this, the possible selections are:

(4 boys out of 5) * (2 girls out of 2) + (5 boys out of 5) * (1 girl out of 2)

= 5C4 * 2C2 + 5C5 * 2C1

So for some **general values** of **P**, **Q** and **N**, the approach can be visualised as:

where

Follow the steps mentioned below to implementation it:

- Start iterating a loop from
**i = 4**till**i = P**. - At each iteration calculate the number of possible ways if we choose
**i boys**and**(N-i) girls,**using combination

**Add**the possible value for each iteration as the**total**number of ways.- Return the total calculated ways at the end.

Below is the implementation of the approach:

## C++

`#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to calculate factorial` `long` `long` `int` `fact(` `int` `f)` `{` ` ` `f++;` ` ` `long` `long` `int` `ans = 1;` ` ` `// Loop to calculate factorial of f` ` ` `while` `(--f > 0)` ` ` `ans = ans * f;` ` ` `return` `ans;` `}` `// Function to calculate combinaion nCr` `long` `long` `int` `ncr(` `int` `n, ` `int` `r)` `{` ` ` `return` `(fact(n) / (fact(r) * fact(n - r)));` `}` `// Function to calculate the number of ways` `long` `long` `int` `countWays(` `int` `n, ` `int` `p, ` `int` `q)` `{` ` ` `long` `long` `int` `sum = 0;` ` ` `// Loop to calculate the number of ways` ` ` `for` `(` `long` `long` `int` `i = 4; i <= p; i++) {` ` ` `if` `(n - i >= 1 && n - i <= q)` ` ` `sum += (ncr(p, i) * ncr(q, n - i));` ` ` `}` ` ` `return` `sum;` `}` `// Driver code` `int` `main()` `{` ` ` `int` `P = 5, Q = 2, N = 5;` ` ` `cout << countWays(N, P, Q) << endl;` ` ` `return` `0;` `}` |

## Java

`import` `java.util.*;` `class` `GFG{` `// Function to calculate factorial` `static` `int` `fact(` `int` `f)` `{` ` ` `f++;` ` ` `int` `ans = ` `1` `;` ` ` `// Loop to calculate factorial of f` ` ` `while` `(--f > ` `0` `)` ` ` `ans = ans * f;` ` ` `return` `ans;` `}` `// Function to calculate combinaion nCr` `static` `int` `ncr(` `int` `n, ` `int` `r)` `{` ` ` `return` `(fact(n) / (fact(r) * fact(n - r)));` `}` `// Function to calculate the number of ways` `static` `int` `countWays(` `int` `n, ` `int` `p, ` `int` `q)` `{` ` ` `int` `sum = ` `0` `;` ` ` `// Loop to calculate the number of ways` ` ` `for` `(` `int` `i = ` `4` `; i <= p; i++) {` ` ` `if` `(n - i >= ` `1` `&& n - i <= q)` ` ` `sum += (ncr(p, i) * ncr(q, n - i));` ` ` `}` ` ` `return` `sum;` `}` `// Driver code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `P = ` `5` `, Q = ` `2` `, N = ` `5` `;` ` ` `System.out.print(countWays(N, P, Q) +` `"\n"` `);` `}` `}` `// This code is contributed by 29AjayKumar` |

## Python3

`# Function to calculate factorial` `def` `fact(f):` ` ` `ans ` `=` `1` ` ` `# Loop to calculate factorial of f` ` ` `while` `(f):` ` ` `ans ` `=` `ans ` `*` `f` ` ` `f ` `-` `=` `1` ` ` `return` `ans` `# Function to calculate combinaion nCr` `def` `ncr(n, r):` ` ` `return` `(fact(n) ` `/` `(fact(r) ` `*` `fact(n ` `-` `r)))` `# Function to calculate the number of ways` `def` `countWays(n, p, q):` ` ` `sum` `=` `0` ` ` `# Loop to calculate the number of ways` ` ` `for` `i ` `in` `range` `(` `4` `, p ` `+` `1` `):` ` ` `if` `(n ` `-` `i >` `=` `1` `and` `n ` `-` `i <` `=` `q):` ` ` `sum` `+` `=` `(ncr(p, i) ` `*` `ncr(q, n ` `-` `i))` ` ` `return` `(` `int` `)(` `sum` `)` `# Driver code` `P ` `=` `5` `Q ` `=` `2` `N ` `=` `5` `print` `(countWays(N, P, Q))` ` ` `# This code is contributed by gfgking.` |

## C#

`// C# program for the above approach` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG {` `// Function to calculate factorial` `static` `int` `fact(` `int` `f)` `{` ` ` `f++;` ` ` `int` `ans = 1;` ` ` `// Loop to calculate factorial of f` ` ` `while` `(--f > 0)` ` ` `ans = ans * f;` ` ` `return` `ans;` `}` `// Function to calculate combinaion nCr` `static` `int` `ncr(` `int` `n, ` `int` `r)` `{` ` ` `return` `(fact(n) / (fact(r) * fact(n - r)));` `}` `// Function to calculate the number of ways` `static` `int` `countWays(` `int` `n, ` `int` `p, ` `int` `q)` `{` ` ` `int` `sum = 0;` ` ` `// Loop to calculate the number of ways` ` ` `for` `(` `int` `i = 4; i <= p; i++) {` ` ` `if` `(n - i >= 1 && n - i <= q)` ` ` `sum += (ncr(p, i) * ncr(q, n - i));` ` ` `}` ` ` `return` `sum;` `}` ` ` `// Driver Code` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` `int` `P = 5, Q = 2, N = 5;` ` ` `Console.Write(countWays(N, P, Q));` ` ` `}` `}` `// This code is contributed by sanjoy_62.` |

## Javascript

`<script>` ` ` `// Function to calculate factorial` ` ` `const fact = (f) => {` ` ` `f++;` ` ` `let ans = 1;` ` ` `// Loop to calculate factorial of f` ` ` `while` `(--f > 0)` ` ` `ans = ans * f;` ` ` `return` `ans;` ` ` `}` ` ` `// Function to calculate combinaion nCr` ` ` `const ncr = (n, r) => {` ` ` `return` `(fact(n) / (fact(r) * fact(n - r)));` ` ` `}` ` ` `// Function to calculate the number of ways` ` ` `const countWays = (n, p, q) => {` ` ` `let sum = 0;` ` ` `// Loop to calculate the number of ways` ` ` `for` `(let i = 4; i <= p; i++) {` ` ` `if` `(n - i >= 1 && n - i <= q)` ` ` `sum += (ncr(p, i) * ncr(q, n - i));` ` ` `}` ` ` `return` `sum;` ` ` `}` ` ` `// Driver code` ` ` `let P = 5, Q = 2, N = 5;` ` ` `document.write(countWays(N, P, Q));` ` ` `// This code is contributed by rakeshsahni` `</script>` |

**Output**

10

**Time Complexity:** O(N^{2})**Auxiliary Space:** O(1)