Question

3) we have 2 engineers and 10 computer scientists. we need to create a committee of 10 people with at least one engineer represented there. how many different ways are there to create such a committee? show at least 2 different ways to solve this problem.

Answer 1

We can solve the committee formation problem using the exclude and include method or the direct method, both yielding 65 different ways to create a committee with at least one engineer.

Step-by-step explanation:

To solve the problem of forming a committee of 10 people with at least one engineer, we can use combinations in two different ways:

1. Exclude and Include Method: We first calculate the total number of ways to form a committee without any restrictions, which is choosing 10 people out of 12, and then subtract the number of ways a committee can be formed with zero engineers, which is choosing 10 computer scientists out of 10.
2. Direct Method: Alternatively, we include at least one engineer in the committee by picking one engineer in 2 ways and then choosing 9 more members from the remaining 10 computer scientists. Since there must be at least one engineer, we could also pick both engineers and then choose the remaining 8 members from the computer scientists.

Now we compute these methods numerically:

1. Total ways with no restriction: C(12, 10) = 66 ways
   Ways with no engineers: C(10, 10) = 1 way
   Ways with at least one engineer: 66 - 1 = 65 ways

2. Ways with one engineer: 2 * C(10, 9) = 2 * 10 = 20 ways
   Ways with two engineers: C(2, 2) * C(10, 8) = 1 * 45 = 45 ways
   Total ways with at least one engineer: 20 + 45 = 65 ways

Both methods yield the same result: 65 different ways to create a committee with at least one engineer.

Answer 2

Step-by-step explanation:

Method 1: Counting the number of committees with exactly one engineer:

There are 2 ways to choose the engineer, and 10 ways to choose the remaining 9 computer scientists. Hence, there are 2 * 10 = 20 different committees with exactly one engineer.

Method 2: Using the combinations formula:

There are 12 people in total, and we need to choose 10 of them for the committee. We can use the combinations formula to calculate the number of different ways to choose 10 people from 12: C(12, 10) = (12!) / (10! * (12-10)!) = (12!) / (10! * 2!) = 66.

So, there are 66 different ways to create a committee of 10 people with at least one engineer represented.