A committee of 5 people is to be selected from 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 woman?

Options

A. `1/9`
B. `(240)/(1,001)`
C. `1/3`
D. `(1,260)/(3,003)`

Explanation

To solve the problem of finding the probability that a committee of 5 people consists of 3 men and 2 women selected from a pool of 6 men and 9 women, we can follow these steps:

### Step 1: Determine the total number of ways to select 5 people from 15.
The total number of people available is 6 men + 9 women = 15 people. The number of ways to choose 5 people from 15 is given by the combination formula \( \binom{n}{r} \), which is calculated as:

\[ \text{Total ways} = \binom{15}{5} = \frac{15!}{5!(15-5)!} = \frac{15!}{5! \cdot 10!} \]

### Step 2: Calculate \( \binom{15}{5} \).
Now we can compute \( \binom{15}{5} \):

\[ \binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = \frac{360360}{120} = 3003 \]

### Step 3: Determine the number of ways to select 3 men from 6 and 2 women from 9.
Next, we calculate the number of ways to select 3 men from 6 and 2 women from 9. This can be expressed as:

\[ \text{Ways to select 3 men} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} \]

Calculating \( \binom{6}{3} \):

\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

Now for the women:

\[ \text{Ways to select 2 women} = \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9!}{2! \cdot 7!} \]

Calculating \( \binom{9}{2} \):

\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 4: Calculate the total number of favorable outcomes.
Now, we multiply the number of ways to select the men and the women:

\[ \text{Total favorable outcomes} = \binom{6}{3} \times \binom{9}{2} = 20 \times 36 = 720 \]

### Step 5: Calculate the probability.
Finally, the probability that the committee consists of 3 men and 2 women is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{720}{3003} \]

### Step 6: Simplify the probability.
Now, we simplify \( \frac{720}{3003} \):

\[ \text{Probability} = \frac{720}{3003} \approx 0.23976 \]

### Final Result:
Thus, the probability that the committee consists of 3 men and 2 women is \( \frac{720}{3003} \).