# 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:
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

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

Algorithm:

• Create a method named “fact” that takes a parameter n and returns the factorial of n.
• Create a variable named “fact” and initialize it to 1.
• Start a for loop and traverse through the range 2 to n (inclusive) and multiply each number with the “fact” variable.
•  Return the value of “fact”.
• Create another method named “ncr” that takes two parameters n and r, and returns the value of nCr.
• Inside the “ncr” function, use the “fact” function to calculate the factorial of n, r, and n-r.
• Calculate nCr using the formula n! / (r! * (n-r)!).
• Return the value of nCr
• Create another method “ways” that takes four parameters m, w, n, and k, and returns the total possible ways.
• Create a variable named “ans” and initialize it to 0.
• Start a while loop to iterate until m is greater than or equal to k.
• Inside the while loop, add the value of nCr(m, k) multiplied by nCr(w, n-k) to the “ans” variable.
•  Increment k by 1 in each iteration.
•  Return the value of “ans”.

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`

## Javascript

 ``

Output:

`756`

Time Complexity: O(n)
Auxiliary Space: O(1)

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

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