A committee of five is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is

Options

A. `(2)/(3)`
B. `(4)/(9)`
C. `(1)/(2)`
D. `(5)/(9)`

Explanation

To solve the problem of finding the probability that a certain married couple will either serve together or not at all on a committee of five chosen from a group of nine people, we can follow these steps:

### Step 1: Determine the Total Number of Ways to Choose the Committee
We need to find the total number of ways to choose a committee of 5 from 9 people. This can be calculated using the combination formula:

\[ \text{Total ways} = \binom{9}{5} \]

### Step 2: Calculate the Total Ways
Using the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

We can calculate:

\[ \binom{9}{5} = \frac{9!}{5! \cdot (9-5)!} = \frac{9!}{5! \cdot 4!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 126 \]

### Step 3: Calculate the Favorable Outcomes
We need to consider two cases for the married couple:

1. **Case 1**: The couple is included in the committee.
- If the couple is included, we have already chosen 2 members (the couple), and we need to choose 3 more members from the remaining 7 people:
\[ \text{Ways with couple} = \binom{7}{3} \]

2. **Case 2**: The couple is not included in the committee.
- If the couple is not included, we need to choose all 5 members from the remaining 7 people:
\[ \text{Ways without couple} = \binom{7}{5} \]

### Step 4: Calculate the Combinations for Each Case
Calculating the combinations:

- For Case 1:
\[ \binom{7}{3} = \frac{7!}{3! \cdot (7-3)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

- For Case 2:
\[ \binom{7}{5} = \binom{7}{2} = \frac{7!}{2! \cdot (7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 5: Total Favorable Outcomes
Now, we can add the favorable outcomes from both cases:

\[ \text{Total favorable outcomes} = \binom{7}{3} + \binom{7}{5} = 21 + 21 = 42 \]

### Step 6: Calculate the Probability
Finally, the probability that the couple will either serve together or not at all is given by the ratio of favorable outcomes to total outcomes:

\[ P = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{42}{126} = \frac{1}{3} \]

### Final Answer
Thus, the probability that the certain married couple will either serve together or not at all is:

\[ \frac{1}{3} \]