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

Answer

To solve the problem of finding the probability that a certain married couple will either serve together or not at all in a committee of five chosen from a group of 9 people, we can follow these steps:

### Step 1: Understand the Total Number of Ways to Choose the Committee
First, we need to determine the total number of ways to choose a committee of 5 from 9 people. This can be calculated using the combination formula:

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

where \( n \) is the total number of people and \( r \) is the number of people to choose.

For our case:
\[ \text{Total ways} = \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = 126 \]

### Step 2: Calculate the Number of Favorable Outcomes
Next, we need to calculate the number of favorable outcomes where the married couple either serves together or not at all.

#### Case 1: The Couple Serves Together
If the couple serves together, we can treat them as a single unit or "block." This means we now have 8 units to choose from (the couple as one unit and the other 7 individuals).

We need to choose 4 more members from the remaining 7 people:
\[ \text{Ways when couple serves together} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = 35 \]

#### Case 2: The Couple Does Not Serve at All
If the couple does not serve at all, we need to choose all 5 members from the remaining 7 people:
\[ \text{Ways when couple does not serve} = \binom{7}{5} = \frac{7!}{5!(7-5)!} = 21 \]

### Step 3: Combine the Favorable Outcomes
Now, we can combine the two cases to find the total number of favorable outcomes:
\[ \text{Total favorable outcomes} = \text{Ways when couple serves together} + \text{Ways when couple does not serve} = 35 + 21 = 56 \]

### Step 4: Calculate the Probability
Finally, we can calculate the probability that the couple will either serve together or not at all:
\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total ways}} = \frac{56}{126} = \frac{4}{9} \]

### Final Answer
The probability that the married couple will either serve together or not at all is \( \frac{4}{9} \).

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