A school has 8 math teachers, 6 English teachers, and 2 geography teachers. From this group, a 5 teacher committee is required. Calculate the number of ways that this committee can be formed if at least one geography teachers must be on the committee.

Question

A school has 8 math teachers, 6 English teachers, and 2 geography teachers. From this group, a 5 teacher committee is required. Calculate the number of ways that this committee can be formed if at least one geography teachers must be on the committee.

Answer 1

The number of committees without restrictions is C(16,5) = 4368

The number of committees without any geography teachers is C(14,5) = 2002

So the number of ways with at least one geography teacher is 4368-2002 = 2366

or:
1 geography teacher : C(2,1)*C(14,4) = 2002
2 geography teachers : C(2,2)*C(14,3) = 364

sum = 2002+364 = 2366 (as above)

Answer 2

To calculate the number of ways that this committee can be formed, we need to consider the different scenarios in which at least one geography teacher is selected.

Let's break it down step by step:

1. Calculate the total number of ways to select a 5-teacher committee from the given group. As the committee can consist of math, English, and geography teachers, we will consider all possible combinations of teachers without any restrictions. Use the combination formula:

C(n, r) = n! / (r!(n - r)!)

where n is the total number of teachers and r is the number of teachers required for the committee.

In this case, we have a total of 8 math teachers, 6 English teachers, and 2 geography teachers to choose from, so n = 8 + 6 + 2 = 16 and r = 5.

C(16, 5) = 16! / (5!(16 - 5)!)
= 4368

There are 4368 ways to select a 5-teacher committee from the given group without any restrictions.

2. Now let's consider the scenario where no geography teacher is selected. In this case, we need to calculate the number of ways to select a 5-teacher committee from the remaining math and English teachers only.

We have a total of 8 math teachers and 6 English teachers to choose from, so n = 8 + 6 = 14 and r = 5.

C(14, 5) = 14! / (5!(14 - 5)!)
= 2002

There are 2002 ways to select a 5-teacher committee without any geography teacher.

3. Finally, let's subtract the number of committees without a geography teacher from the total number of committees to find the number of committees with at least one geography teacher.

Number of committees with at least one geography teacher = Total committees - Committees without a geography teacher

= 4368 - 2002

= 2366

Therefore, there are 2366 ways to form a 5-teacher committee from the given group if at least one geography teacher must be on the committee.