A committee of three people is selected at random from a set consisting of five teachers, six parents of students, and
three alumni.
a. What is the probability of the event that the committee consists of all teachers?
b. What is the probability of the event that the committee has no teachers?
a. The probability the committee consists of all teachers is
(Type an integer or a simplified fraction.)

Question

UpStudy AI · Accepted Answer

To find the probability that the committee consists of all teachers:

1. **Total number of ways to select 3 people from 14:**
   $$
   \binom{14}{3} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364
   $$

2. **Number of ways to select 3 teachers from 5:**
   $$
   \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
   $$

3. **Probability:**
   $$
   \text{Probability} = \frac{\binom{5}{3}}{\binom{14}{3}} = \frac{10}{364} = \frac{5}{182}
   $$

**Answer:**  
Problem a Answer:

The probability is $\dfrac{5}{182}$.
The probability that the committee consists of all teachers is $\dfrac{5}{182}$.

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