Question

A team of 5 must be selected from 4 men and 4 women. At least 2 men and 2 women must be included. X is the total number of valid teams possible if A and B (both women) refuse to work together. Y is the total number of valid teams possible when C (a man) must be included, and D (a woman) is included. Find the values of X and Y.
Options for X: \{30, 32\}
Options for Y: \{18, 24, 30, 36, 48\}

Hint:

For combinatorics problems with "at least" or "at most" conditions, break the problem down into distinct cases that satisfy the condition. For "refuse to work together" (or "not together") conditions, it's almost always easier to use the complement method: calculate the total and subtract the cases where they are together.

Answer 1

Step 1: Determine Possible Team Compositions
The team has 5 members. It must contain at least 2 men (M) and 2 women (W). The possible compositions are:

Case 1: 2 Men and 3 Women (2M, 3W)
Case 2: 3 Men and 2 Women (3M, 2W)
Step 2: Calculate X
X is the number of teams where women A and B do not work together. The easiest way is to calculate the total number of valid teams and subtract the number of teams where A and B are together.

Total Valid Teams (no restrictions):

Ways for (2M, 3W) = \( \binom{4}{2} \times \binom{4}{3} = 6 \times 4 = 24 \)
Ways for (3M, 2W) = \( \binom{4}{3} \times \binom{4}{2} = 4 \times 6 = 24 \)
Total = \(24 + 24 = 48\) teams.

Teams where A and B are TOGETHER: We must include A and B. This means we have already selected 2 women. We need to select 3 more people to complete the team of 5.

To form a (2M, 3W) team: We have 2W (A and B). We need 2M and 1W more. Select 2M from 4 men, and 1W from the remaining 2 women. \[ \binom{4}{2} \times \binom{2}{1} = 6 \times 2 = 12 \]
To form a (3M, 2W) team: We have 2W (A and B). We need 3M more. Select 3M from 4 men. \[ \binom{4}{3} \times \binom{2}{0} = 4 \times 1 = 4 \]
Total teams with A and B together = \(12 + 4 = 16\).

Calculate X: \[ X = (\text{Total Valid Teams}) - (\text{Teams with A and B together}) = 48 - 16 = 32 \]
Step 3: Calculate Y
Y is the number of teams where man C and woman D are included. We must include C and D. We have already selected 1 man and 1 woman. We need to select 3 more people from the remaining 3 men and 3 women to complete the team of 5. The final team must satisfy the (at least 2M, 2W) condition.

To form a final team of (2M, 3W): We have (1M, 1W). We need to select 1 more man and 2 more women from the remaining (3M, 3W). \[ \text{Ways} = \binom{3}{1} \times \binom{3}{2} = 3 \times 3 = 9 \]
To form a final team of (3M, 2W): We have (1M, 1W). We need to select 2 more men and 1 more woman from the remaining (3M, 3W). \[ \text{Ways} = \binom{3}{2} \times \binom{3}{1} = 3 \times 3 = 9 \]
Calculate Y: \[ Y = (\text{Ways for 2M, 3W}) + (\text{Ways for 3M, 2W}) = 9 + 9 = 18 \]
Step 4: Final Answer
The values are X = 32 and Y = 18.