Question

A committee of 4 people is selected from the members of a school council which consists of 5 students, 4 teachers and 3 administrators. The probability that the committee has no teachers is

• A. \(\frac{13}{99}\)
• B. \(\frac{14}{99}\)
• C. \(\frac{16}{99}\)
• D. \(\frac{20}{99}\)
• E. \(\frac{25}{99}\)

Hint:

Instead of calculating factorials fully, simplify the fractions in combination notation. For example, \(\frac{8 \cdot 7 \cdot 6 \cdot 5}{12 \cdot 11 \cdot 10 \cdot 9}\) allows you to cancel terms much faster during the exam.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a probability problem based on combinations. Probability is defined as the number of favorable outcomes divided by the total number of outcomes.

Step 2: Key Formula or Approach:
1. Total selection: Select 4 from \(5+4+3=12\) people.
2. Favorable selection: Select 4 from \(5 \text{ (students)} + 3 \text{ (administrators)} = 8\) non-teacher people.
3. Combination formula: \(^nC_r = \frac{n!}{r!(n-r)!}\).

Step 3: Detailed Explanation:
Total number of ways to select 4 out of 12: \[ ^{12}C_4 = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 11 \times 5 \times 9 = 495 \]
Number of ways to select 4 out of 8 non-teachers: \[ ^8C_4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 2 \times 7 \times 5 = 70 \]
Probability \(P\): \[ P = \frac{70}{495} = \frac{14}{99} \]

Step 4: Final Answer:
The probability is \(\frac{14}{99}\).