A husband and wife appear in an interview for two vacancies in the same post .The probability of husband 's selection is 1/7 and that of wife's 1/5
What is the probability that at least one of them will selected ?

Options

A. `(2)/(7)`
B. `(1)/(35)`
C. `(24)/(35)`
D. `(11)/(35)`

Answer

To solve the problem of finding the probability that at least one of the husband or wife will be selected for the vacancies, we can follow these steps:

### Step 1: Identify the probabilities of selection
Let:
- \( P(A) \) = Probability of husband being selected = \( \frac{1}{7} \)
- \( P(B) \) = Probability of wife being selected = \( \frac{1}{5} \)

### Step 2: Calculate the probabilities of non-selection
Now, we need to find the probabilities that each of them is not selected:
- Probability that the husband is not selected:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{7} = \frac{6}{7} \]
- Probability that the wife is not selected:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{5} = \frac{4}{5} \]

### Step 3: Calculate the probability that neither is selected
The probability that neither the husband nor the wife is selected can be found by multiplying their individual probabilities of non-selection:
\[ P(A' \cap B') = P(A') \times P(B') = \frac{6}{7} \times \frac{4}{5} \]
Calculating this gives:
\[ P(A' \cap B') = \frac{6 \times 4}{7 \times 5} = \frac{24}{35} \]

### Step 4: Calculate the probability that at least one is selected
The probability that at least one of them is selected is the complement of the probability that neither is selected:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - \frac{24}{35} \]
Calculating this gives:
\[ P(A \cup B) = \frac{35 - 24}{35} = \frac{11}{35} \]

### Final Answer
Thus, the probability that at least one of them will be selected is:
\[ \frac{11}{35} \]