A hotel has five single room available, for which six men and three women apply.
What is the probability that the rooms will be rented to three men and two women ?

Options

A. `23/112`
B. `97/251`
C. `10/21`
D. `5/9`

Explanation

To solve the problem, we need to find the probability that the rooms will be rented to three men and two women from a total of six men and three women who applied for the five available rooms.

### Step-by-Step Solution:

1. **Identify Total Applicants**:
- Total number of applicants = 6 men + 3 women = 9 applicants.

2. **Total Rooms Available**:
- Total rooms available = 5 rooms.

3. **Total Ways to Fill the Rooms**:
- The total number of ways to choose 5 applicants from 9 is given by the combination formula \( \binom{n}{r} \), which represents the number of ways to choose \( r \) successes in \( n \) trials.
- So, the total ways to fill the rooms is \( \binom{9}{5} \).

4. **Favorable Outcomes**:
- We need to find the number of ways to choose 3 men from 6 and 2 women from 3.
- The number of ways to choose 3 men from 6 is \( \binom{6}{3} \).
- The number of ways to choose 2 women from 3 is \( \binom{3}{2} \).
- Therefore, the total number of favorable outcomes is \( \binom{6}{3} \times \binom{3}{2} \).

5. **Calculate Combinations**:
- Calculate \( \binom{9}{5} \):
\[ \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 126 \]
- Calculate \( \binom{6}{3} \):
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
- Calculate \( \binom{3}{2} \):
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = 3 \]

6. **Total Favorable Outcomes**:
- Now, multiply the combinations:
\[ \text{Favorable outcomes} = \binom{6}{3} \times \binom{3}{2} = 20 \times 3 = 60 \]

7. **Calculate Probability**:
- The probability \( P \) that the rooms will be rented to three men and two women is given by:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{60}{126} \]
- Simplifying \( \frac{60}{126} \):
\[ P = \frac{10}{21} \]

### Final Answer:
The probability that the rooms will be rented to three men and two women is \( \frac{10}{21} \).