# Ways of selecting men and women from a group to make a team

Given four integers n, w, m and k where,

• m is the total number of men.
• w is the total number of women.
• n is the total number of people that need to be selected to form the team.
• k is the minimum number of men that have to be selected.

The task is to find the number of ways in which the team can be formed.

Examples:

Input: m = 2, w = 2, n = 3, k = 1
Output: 4
There are 2 men, 2 women. We need to make a team of size 3 with at least one man and one woman. We can make the team in following ways.
m1 m2 w1
m1 w1 w2
m2 w1 w2
m1 m2 w2

Input: m = 7, w = 6, n = 5, k = 3
Output: 756

Input: m = 5, w = 6, n = 6, k = 3
Output: 281

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: Since, we have to take at least k men.

Totals ways = Ways when ‘k’ men are selected + Ways when ‘k+1’ men are selected + … + when ‘n’ men are selected

.
Taking the first example from above where out of 7 men and 6 women, total 5 people need to be selected with at least 3 men,
Number of ways = (7C3 x 6C2) + (7C4 x 6C1) + (7C5)
= 7 x 6 x 5 x 6 x 5 + (7C3 x 6C1) + (7C2)
= 525 + 7 x 6 x 5 x 6 + 7 x 6
= (525 + 210 + 21)
= 756

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Returns factorial ` `// of the number ` `int` `fact(``int` `n) ` `{ ` `    ``int` `fact = 1; ` `    ``for` `(``int` `i = 2; i <= n; i++) ` `        ``fact *= i; ` `    ``return` `fact; ` `} ` ` `  `// Function to calculate ncr ` `int` `ncr(``int` `n, ``int` `r) ` `{ ` `    ``int` `ncr = fact(n) / (fact(r) * fact(n - r)); ` `    ``return` `ncr; ` `} ` ` `  `// Function to calculate ` `// the total possible ways ` `int` `ways(``int` `m, ``int` `w, ``int` `n, ``int` `k) ` `{ ` ` `  `    ``int` `ans = 0; ` `    ``while` `(m >= k) { ` `        ``ans += ncr(m, k) * ncr(w, n - k); ` `        ``k += 1; ` `    ``} ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``int` `m, w, n, k; ` `    ``m = 7; ` `    ``w = 6; ` `    ``n = 5; ` `    ``k = 3; ` `    ``cout << ways(m, w, n, k); ` `} `

## Java

 `// Java implementation of the approach ` ` `  `import` `java.io.*; ` ` `  `class` `GFG { ` ` `  `// Returns factorial ` `// of the number ` `static` `int` `fact(``int` `n) ` `{ ` `    ``int` `fact = ``1``; ` `    ``for` `(``int` `i = ``2``; i <= n; i++) ` `        ``fact *= i; ` `    ``return` `fact; ` `} ` ` `  `// Function to calculate ncr ` `static` `int` `ncr(``int` `n, ``int` `r) ` `{ ` `    ``int` `ncr = fact(n) / (fact(r) * fact(n - r)); ` `    ``return` `ncr; ` `} ` ` `  `// Function to calculate ` `// the total possible ways ` `static` `int` `ways(``int` `m, ``int` `w, ``int` `n, ``int` `k) ` `{ ` ` `  `    ``int` `ans = ``0``; ` `    ``while` `(m >= k) { ` `        ``ans += ncr(m, k) * ncr(w, n - k); ` `        ``k += ``1``; ` `    ``} ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `    ``public` `static` `void` `main (String[] args) { ` `         `  `    ``int` `m, w, n, k; ` `    ``m = ``7``; ` `    ``w = ``6``; ` `    ``n = ``5``; ` `    ``k = ``3``; ` `    ``System.out.println( ways(m, w, n, k)); ` `    ``} ` `} ` `// This Code is contributed ` `// by shs `

## Python3

 `# Python 3 implementation of the approach  ` ` `  `# Returns factorial of the number  ` `def` `fact(n):  ` `    ``fact ``=` `1` `    ``for` `i ``in` `range``(``2``, n ``+` `1``):  ` `        ``fact ``*``=` `i  ` `    ``return` `fact ` ` `  `# Function to calculate ncr  ` `def` `ncr(n, r): ` `    ``ncr ``=` `fact(n) ``/``/` `(fact(r) ``*` `fact(n ``-` `r))  ` `    ``return` `ncr ` ` `  `# Function to calculate  ` `# the total possible ways  ` `def` `ways(m, w, n, k): ` `    ``ans ``=` `0` `    ``while` `(m >``=` `k):  ` `        ``ans ``+``=` `ncr(m, k) ``*` `ncr(w, n ``-` `k)  ` `        ``k ``+``=` `1` ` `  `    ``return` `ans; ` ` `  `# Driver code  ` `m ``=` `7` `w ``=` `6` `n ``=` `5` `k ``=` `3` `print``(ways(m, w, n, k)) ` ` `  `# This code is contributed by sahishelangia `

## C#

 `// C# implementation of the approach ` ` `  `class` `GFG { ` ` `  `// Returns factorial ` `// of the number ` `static` `int` `fact(``int` `n) ` `{ ` `    ``int` `fact = 1; ` `    ``for` `(``int` `i = 2; i <= n; i++) ` `        ``fact *= i; ` `    ``return` `fact; ` `} ` ` `  `// Function to calculate ncr ` `static` `int` `ncr(``int` `n, ``int` `r) ` `{ ` `    ``int` `ncr = fact(n) / (fact(r) * fact(n - r)); ` `    ``return` `ncr; ` `} ` ` `  `// Function to calculate ` `// the total possible ways ` `static` `int` `ways(``int` `m, ``int` `w, ``int` `n, ``int` `k) ` `{ ` ` `  `    ``int` `ans = 0; ` `    ``while` `(m >= k) { ` `        ``ans += ncr(m, k) * ncr(w, n - k); ` `        ``k += 1; ` `    ``} ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `    ``static` `void` `Main () { ` `         `  `    ``int` `m, w, n, k; ` `    ``m = 7; ` `    ``w = 6; ` `    ``n = 5; ` `    ``k = 3; ` `    ``System.Console.WriteLine( ways(m, w, n, k)); ` `    ``} ` `} ` `// This Code is contributed by mits `

## PHP

 `= ``\$k``)  ` `    ``{ ` `        ``\$ans` `+= ncr(``\$m``, ``\$k``) * ` `                ``ncr(``\$w``, ``\$n` `- ``\$k``); ` `        ``\$k` `+= 1; ` `    ``} ` ` `  `    ``return` `\$ans``; ` `} ` ` `  `// Driver code ` `\$m` `= 7; ` `\$w` `= 6; ` `\$n` `= 5; ` `\$k` `= 3; ` `echo` `ways(``\$m``, ``\$w``, ``\$n``, ``\$k``); ` ` `  `// This Code is contributed ` `// by Mukul Singh `

Output:

```756
```

Further Optimization : The above code can be optimized using faster algorithms for binomial coefficient computation.

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.