A committee of 5 persons is to be randomly selected from a group of 5 men and 4 women and a chairperson will be randomly selected from the committee will have exactly 2 women and 3 men and the chairperson will be a man is p, then `(1)/(p)` is equal to

Answer

To solve the problem, we need to determine the probability \( p \) that a randomly selected committee of 5 persons from a group of 5 men and 4 women will have exactly 2 women and 3 men, with the chairperson being a man. We will then find \( \frac{1}{p} \).

### Step 1: Calculate the total number of ways to select a committee of 5 persons from 9 (5 men + 4 women).

The total number of ways to select 5 persons from 9 is given by the combination formula:

\[ \text{Total ways} = \binom{9}{5} = \frac{9!}{5! \cdot (9-5)!} = \frac{9!}{5! \cdot 4!} \]

Calculating this:

\[ \binom{9}{5} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126 \]

### Step 2: Calculate the number of favorable cases (committees with exactly 2 women and 3 men).

To have exactly 2 women and 3 men, we need to choose:

- 2 women from 4
- 3 men from 5

Calculating these combinations:

\[ \text{Ways to choose 2 women} = \binom{4}{2} = \frac{4!}{2! \cdot (4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

\[ \text{Ways to choose 3 men} = \binom{5}{3} = \frac{5!}{3! \cdot (5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Calculate the total number of ways to select a chairperson from the committee.

Since the chairperson must be a man and we have selected 3 men, the number of ways to choose a chairperson is 3.

### Step 4: Calculate the total number of favorable cases.

The total number of favorable cases is given by:

\[ \text{Favorable cases} = (\text{Ways to choose 2 women}) \times (\text{Ways to choose 3 men}) \times (\text{Ways to choose chairperson}) \]

\[ \text{Favorable cases} = 6 \times 10 \times 3 = 180 \]

### Step 5: Calculate the probability \( p \).

The probability \( p \) is given by the ratio of favorable cases to total cases:

\[ p = \frac{\text{Favorable cases}}{\text{Total ways}} = \frac{180}{126} \]

Simplifying this fraction:

\[ p = \frac{30}{21} = \frac{10}{7} \]

### Step 6: Calculate \( \frac{1}{p} \).

Now, we find \( \frac{1}{p} \):

\[ \frac{1}{p} = \frac{7}{10} = 0.7 \]

### Final Answer:

Thus, the value of \( \frac{1}{p} \) is \( 0.7 \).

---