Question

A school committee of 4 members is to be formed from 6 teachers and 5 students. If the committee must include at least one student, how many different committees can be formed?

• A. 240
• B. 260
• C. 285
• D. 300
• E. 315

Hint:

For problems asking for "at least one" of something, the indirect method (Total - None) is almost always faster and less prone to calculation errors than calculating each case (exactly 1, exactly 2, etc.) and summing them up.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
This is a problem of combinations, where we need to find the number of ways to select a group of people without regard to the order of selection. The condition "at least one student" is a key constraint. The easiest way to solve "at least one" problems is to calculate the total number of possible combinations and subtract the number of combinations that violate the condition (i.e., having zero students).
Step 2: Key Formula or Approach
The number of combinations of choosing \(k\) items from a set of \(n\) items is given by the formula: \[ C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} \] The approach will be: (Committees with at least one student) = (Total possible committees) - (Committees with no students)
Step 3: Detailed Explanation
Part 1: Calculate the total number of possible committees.
There are a total of \(6 + 5 = 11\) people. We need to choose a committee of 4. \[ \text{Total committees} = C(11, 4) = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!} \] \[ = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 11 \times 10 \times 3 = 330 \] So, there are 330 possible committees in total, without any restrictions.
Part 2: Calculate the number of committees with no students.
This means the committee must be formed entirely of teachers. We need to choose 4 teachers from the available 6 teachers. \[ \text{Committees with no students} = C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6!}{4!2!} \] \[ = \frac{6 \times 5}{2 \times 1} = 15 \] So, there are 15 committees that have zero students.
Part 3: Find the number of committees with at least one student.
\[ \text{Committees with at least one student} = 330 - 15 = 315 \] Step 4: Final Answer
The total number of different committees that can be formed with at least one student is 315. Since this is not one of the options (A, B, C, D), the provided options are incorrect.