In how many ways, a committee of 3 men and 2 women can be formed out of a total of 4 men and 4 women?

Options

A. 15
B. 16
C. 20
D. 24

Answer

To solve the problem of forming a committee of 3 men and 2 women from a total of 4 men and 4 women, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Determine the number of ways to choose 3 men from 4 men.
We need to calculate the number of combinations of choosing 3 men from a total of 4 men. This can be represented mathematically as:

\[ \text{Number of ways to choose 3 men} = \binom{4}{3} \]

### Step 2: Calculate \(\binom{4}{3}\).
Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\):

\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} \]

Calculating the factorials:

\[ = \frac{4 \times 3!}{3! \times 1} = \frac{4}{1} = 4 \]

### Step 3: Determine the number of ways to choose 2 women from 4 women.
Next, we calculate the number of combinations of choosing 2 women from a total of 4 women:

\[ \text{Number of ways to choose 2 women} = \binom{4}{2} \]

### Step 4: Calculate \(\binom{4}{2}\).
Using the combination formula again:

\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} \]

Calculating the factorials:

\[ = \frac{4 \times 3 \times 2!}{2! \times 2!} = \frac{4 \times 3}{2 \times 1} = \frac{12}{2} = 6 \]

### Step 5: Calculate the total number of ways to form the committee.
Now, we multiply the number of ways to choose the men and the women together:

\[ \text{Total ways} = \binom{4}{3} \times \binom{4}{2} = 4 \times 6 = 24 \]

### Final Answer:
Thus, the total number of ways to form a committee of 3 men and 2 women from 4 men and 4 women is **24**.

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