Permutation and combination

Question

Answer 1

We have a total of 15 members in the committee, consisting of 7 men and 8 women.

Now both men and women needs to be assigned for a total of 3 tasks.

These tasks are assigned in such a way that at least both a man and a woman must be assigned a task.

Therefore the cases that can be formed are:

Case 1: 

Task 1 : 1 Man

Task 2 : 1 Man

Task 3 : 1 Woman

Case 2: 

Task 1 : 1 Man

Task 2 : 1 Woman

Task 3 : 1 Woman

[Please note there will be no cases such that we assigned 1 Man to task 1, 1 Woman to task 2 and leave the task 3 as at least a man and a woman are assigned. Every task needs a person.]

All now left is to calculate the separate cases and sum up them.

Case 1: To choose 2 men out of 7 men and 1 woman out of 8 women. 

(7 choose 2) * (8 choose 1) = 21 * 8 = 168

Case 2: To choose 1 men out of 7 men and 2 woman out of 8 woman

(7 choose 1) * (8 choose 2) = 7 * 28 = 196

Therefore total ways = 168 + 196 = 364 ways.

Hence, there are 364 ways to complete the tasks.