To solve the problem of finding the number of committees of three that can be formed consisting of two men and one woman from a group of four men and three women, we will use the concept of combinations in combinatorial mathematics. ### Step-by-Step Explanation 1. **Understanding the Problem**: - We need to form a committee of 3 people: 2 men and 1 woman. - We have 4 men and 3 women to choose from. 2. **Choosing the Men**: - We need to select 2 men from the 4 available men. The number of ways to choose \( r \) items from \( n \) items is given by the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] - For our case, we need to calculate \( \binom{4}{2} \): \[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6 \] - So, there are 6 ways to choose 2 men from 4. 3. **Choosing the Woman**: - Next, we need to select 1 woman from the 3 available women. We will use the same combination formula: - For our case, we need to calculate \( \binom{3}{1} \): \[ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3!}{1! \cdot 2!} = \frac{3}{1} = 3 \] - So, there are 3 ways to choose 1 woman from 3. 4. **Calculating the Total Committees**: - To find the total number of committees, we multiply the number of ways to choose the men by the number of ways to choose the woman: \[ \text{Total Committees} = \binom{4}{2} \times \binom{3}{1} = 6 \times 3 = 18 \] ### Conclusion The total number of committees of three that can be formed consisting of two men and one woman from four men and three women is **18**. ### Explanation of Other Options - **Option A (3)**: This option is incorrect because it underestimates the number of combinations. It does not account for the multiple ways to choose 2 men from 4. - **Option B (6)**: This option only represents the number of ways to choose 2 men but ignores the selection of the woman. - **Option D (24)**: This option incorrectly assumes that the selections are independent and adds extra combinations that do not exist in this context. ### Revision Summary - Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to calculate selections. - Calculate the number of ways to choose men and women separately, then multiply the results. - Ensure to understand the context of the problem to avoid miscalculating combinations. - The final answer for the number of committees is **18**.