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A hotel has five single room available, ...

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 ?

A

`23/112`

B

`97/251`

C

`10/21`

D

`5/9`

Text Solution

AI Generated Solution

The correct Answer is:
To find the probability that the rooms will be rented to three men and two women from a pool of six men and three women, we can follow these steps: ### Step 1: Determine the Total Number of Outcomes We need to find the total number of ways to select 5 people from the 9 applicants (6 men + 3 women). This can be calculated using combinations: \[ \text{Total Outcomes} = \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \] ### Step 2: Determine the Favorable Outcomes for Selecting 3 Men Next, we need to find the number of ways to select 3 men from the 6 men available: \[ \text{Ways to choose 3 men} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \] ### Step 3: Determine the Favorable Outcomes for Selecting 2 Women Now, we find the number of ways to select 2 women from the 3 women available: \[ \text{Ways to choose 2 women} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \] ### Step 4: Calculate the Total Favorable Outcomes The total number of favorable outcomes for the event of selecting 3 men and 2 women is the product of the two combinations calculated above: \[ \text{Favorable Outcomes} = \text{Ways to choose 3 men} \times \text{Ways to choose 2 women} = 20 \times 3 = 60 \] ### Step 5: Calculate the Probability Finally, the probability \( P \) that the rooms will be rented to three men and two women is given by the ratio of favorable outcomes to total outcomes: \[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{60}{126} \] ### Step 6: Simplify the Probability We can simplify the fraction: \[ P = \frac{60 \div 6}{126 \div 6} = \frac{10}{21} \] Thus, the probability that the rooms will be rented to three men and two women is \( \frac{10}{21} \). ---
To find the probability that the rooms will be rented to three men and two women from a pool of six men and three women, we can follow these steps: ### Step 1: Determine the Total Number of Outcomes We need to find the total number of ways to select 5 people from the 9 applicants (6 men + 3 women). This can be calculated using combinations: \[ \text{Total Outcomes} = \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \] ...
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