Four couples (husband and wife) decide to form a committee of four members. Find the number of different committees that can be formed in which no couple finds a place. A)

Options

A. A. 8
B. B. 12
C. C. 16
D. D. none of these

Answer

To solve the problem of forming a committee of four members from four couples (husband and wife) such that no couple is included in the committee, we can break down the solution step by step.

### Step 1: Understand the total number of people
We have 4 couples, which means there are a total of 8 people (4 husbands and 4 wives).

### Step 2: Determine the committee composition
We need to form a committee of 4 members. Since no couple can be together, we can have the following combinations:
1. 4 men
2. 3 men and 1 woman
3. 2 men and 2 women
4. 1 man and 3 women
5. 0 men and 4 women

### Step 3: Calculate the number of ways for each combination

#### Case 1: 4 Men
- We can choose all 4 men from the 4 available men.
- Number of ways = \( \binom{4}{4} = 1 \)

#### Case 2: 3 Men and 1 Woman
- Choose 3 men from 4: \( \binom{4}{3} = 4 \)
- Choose 1 woman from the remaining 4 women (since the wife of the chosen men cannot be selected): \( \binom{4}{1} = 4 \)
- Total for this case = \( 4 \times 4 = 16 \)

#### Case 3: 2 Men and 2 Women
- Choose 2 men from 4: \( \binom{4}{2} = 6 \)
- Choose 2 women from the remaining 2 women (the wives of the chosen men cannot be selected): \( \binom{2}{2} = 1 \)
- Total for this case = \( 6 \times 1 = 6 \)

#### Case 4: 1 Man and 3 Women
- Choose 1 man from 4: \( \binom{4}{1} = 4 \)
- Choose 3 women from the remaining 3 women (the wife of the chosen man cannot be selected): \( \binom{3}{3} = 1 \)
- Total for this case = \( 4 \times 1 = 4 \)

#### Case 5: 0 Men and 4 Women
- This case is not possible since we need to form a committee of 4 members, and we cannot select 4 women without including their husbands.

### Step 4: Add all the cases together
Now, we sum the total number of ways from all the valid cases:
- Case 1: 1
- Case 2: 16
- Case 3: 6
- Case 4: 4
- Case 5: 0

Total = \( 1 + 16 + 6 + 4 = 27 \)

### Final Answer
The total number of different committees that can be formed in which no couple finds a place is **27**.

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