Question

7. A committee of 6 people is to be selected from a pool of 10 engineers and 8 designers. In how many ways can this committee be formed, considering that: a) At least 3 engineers must be included? b) No more than 4 designers are included?​

Answer 1

To form the committee, we need to consider the number of engineers and designers included. The number of ways to form the committee with at least 3 engineers is (10 choose 3) * ((10+8) choose 3). The number of ways to form the committee with no more than 4 designers is (8 choose 1) * (10 choose (6-1)) + (8 choose 2) * (10 choose (6-2)) + (8 choose 3) * (10 choose (6-3)) + (8 choose 4) * (10 choose (6-4)).

Step-by-step explanation:

To find the number of ways a committee can be formed, we need to consider the number of engineers and designers included. We will break this down into two parts:

a) At least 3 engineers must be included:

In this case, we need to select 3 engineers from the pool of 10 engineers. The remaining 3 members can be either engineers or designers.

So, we have 10 choose 3 ways to select the engineers and (10+8) choose 3 ways to select the remaining members.

Therefore, the total number of ways to form the committee with at least 3 engineers is (10 choose 3) * ((10+8) choose 3).

b) No more than 4 designers are included:

In this case, we need to select 1, 2, 3, or 4 designers from the pool of 8 designers. The remaining members can be engineers.

So, we have 8 choose 1 + 8 choose 2 + 8 choose 3 + 8 choose 4 ways to select the designers and 10 choose (6-1), 10 choose (6-2), 10 choose (6-3), or 10 choose (6-4) ways to select the remaining members.

Therefore, the total number of ways to form the committee with no more than 4 designers is (8 choose 1) * (10 choose (6-1)) + (8 choose 2) * (10 choose (6-2)) + (8 choose 3) * (10 choose (6-3)) + (8 choose 4) * (10 choose (6-4)).