A husband and a wife appear in a interview for two vacancies in the same post. The probability of husband's selection is `(1)/(7)` and that of wife's is `(1)/(6)`. What is the probability that none of them will be selected.

Options

A. `(2)/(13)`
B. `(5)/(7)`
C. `(1)/(42)`
D. `(41)/(42)`

Answer

To solve the problem, we need to calculate the probability that neither the husband nor the wife will be selected for the vacancies.

### Step-by-Step Solution:

1. **Identify the probabilities of selection:**
- The probability of the husband being selected, \( P(H) = \frac{1}{7} \).
- The probability of the wife being selected, \( P(W) = \frac{1}{6} \).

2. **Calculate the probabilities of not being selected:**
- The probability of the husband not being selected, \( P(H') = 1 - P(H) = 1 - \frac{1}{7} = \frac{6}{7} \).
- The probability of the wife not being selected, \( P(W') = 1 - P(W) = 1 - \frac{1}{6} = \frac{5}{6} \).

3. **Since the selections are independent events, we can multiply the probabilities of not being selected:**
- The probability that neither the husband nor the wife is selected is given by:
\[ P(H' \cap W') = P(H') \times P(W') = \frac{6}{7} \times \frac{5}{6} \]

4. **Perform the multiplication:**
- Simplifying \( \frac{6}{7} \times \frac{5}{6} \):
\[ P(H' \cap W') = \frac{6 \times 5}{7 \times 6} = \frac{30}{42} \]
- Simplifying \( \frac{30}{42} \):
\[ P(H' \cap W') = \frac{5}{7} \]

5. **Final Result:**
- The probability that neither the husband nor the wife will be selected is \( \frac{5}{7} \).