To solve the problem of how many ways a delegation of 3 can be chosen from among 5 men and 3 women, with the condition that at least one man and at least one woman must be included, we can use combinatorial counting methods. Let's break this down step-by-step. ### Step 1: Total Combinations Without Restrictions First, we calculate the total number of ways to choose 3 people from the 8 available (5 men + 3 women) without any restrictions. This can be done using the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial. In our case, \( n = 8 \) and \( r = 3 \): \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \] So, there are 56 ways to choose any 3 people from the group of 8. ### Step 2: Exclude Invalid Combinations Next, we need to exclude the combinations that do not meet the requirement of having at least one man and one woman. There are two cases to consider: 1. **All Men**: Choosing 3 men from the 5 available. 2. **All Women**: Choosing 3 women from the 3 available. #### Case 1: All Men The number of ways to choose 3 men from 5 is: \[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \] #### Case 2: All Women The number of ways to choose 3 women from 3 is: \[ \binom{3}{3} = 1 \] ### Step 3: Total Invalid Combinations Now, we add the invalid combinations together: \[ \text{Total Invalid} = \text{All Men} + \text{All Women} = 10 + 1 = 11 \] ### Step 4: Valid Combinations To find the valid combinations that include at least one man and one woman, we subtract the invalid combinations from the total combinations: \[ \text{Valid Combinations} = \text{Total Combinations} - \text{Total Invalid} = 56 - 11 = 45 \] ### Conclusion Thus, the number of ways to choose a delegation of 3 that includes at least one man and one woman is **45**. ### Summary of Options - **A. 15**: Too low; does not account for the combinations correctly. - **B. 28**: Incorrect; does not consider all valid combinations. - **C. 30**: Also incorrect; undercounts the valid combinations. - **D. 45**: Correct; accurately reflects the valid combinations. ### Revision Summary - Use the combination formula \(\binom{n}{r}\) to calculate total combinations. - Exclude invalid combinations that do not meet the criteria. - Subtract invalid combinations from the total to find valid options. - Always check each option against your final answer to ensure accuracy.