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.
Input: m = 2, w = 2, n = 3, k = 1
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
Input: m = 5, w = 6, n = 6, k = 3
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)
Below is the implementation of the above approach:
Further Optimization : The above code can be optimized using faster algorithms for binomial coefficient computation.
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