Question

Four women and three men appear for an interview. Three persons have to be selected. Find the probability of selecting one man and two women in the team.

• A. 13/25
• B. 18/35
• C. 19/45
• D. 9/25

Hint:

Always calculate the denominator first in probability questions involving combinations.
Finding that the total ways is 35 immediately helps you narrow down the options, as the correct answer will likely have a denominator that is a factor of 35 (like 35 or 5). Option (B) stands out immediately.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The pool of candidates consists of 4 women and 3 men, totaling 7 candidates.
We need to select 3 persons from this pool.
We are asked to find the probability that this selection consists of exactly 1 man and 2 women.

Step 2: Key Formula or Approach:
We use the classical definition of probability:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
The number of ways to select \(r\) items from a pool of \(n\) items is calculated using the combinations formula:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]

Step 3: Detailed Explanation:
$\bullet$

Step 1: Calculate the Total Number of Possible Outcomes:
We need to select 3 persons out of the total 7 candidates (4 women + 3 men).
The total number of ways to do this is:
\[ \text{Total Outcomes} = \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \]
\[ \text{Total Outcomes} = 7 \times 5 = 35 \]
$\bullet$

Step 2: Calculate the Number of Favorable Outcomes:
We need to select exactly 1 man and 2 women.
- Number of ways to select 1 man from the 3 available men:
\[ \binom{3}{1} = 3 \]
- Number of ways to select 2 women from the 4 available women:
\[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \]
Using the multiplication principle, the total number of favorable ways is:
\[ \text{Favorable Outcomes} = \binom{3}{1} \times \binom{4}{2} = 3 \times 6 = 18 \]
$\bullet$

Step 3: Calculate the Probability:
Now, substitute the values back into the probability formula:
\[ P(\text{Selecting 1 man and 2 women}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{18}{35} \]

Step 4: Final Answer:
The probability of selecting one man and two women is 18/35.