How Many Groups Can Be Formed from 8 Men and 5 Women?

The problem of forming groups with equal numbers of men and women can be approached systematically by breaking it down into several steps. This article explores the methodology to solve such a problem, ensuring that every combination is accounted for while adhering to the given constraints.

Determining Possible Sizes of Groups

The maximum number of men in a group is 5 because there are only 5 women in the pool. Therefore, we can form groups of sizes from 1 to 5, with the number of men equaling the number of women. Let's break down this process step by step.

Step 1: Identify Possible Group Sizes

The possible group sizes are as follows:

1 man and 1 woman 2 men and 2 women 3 men and 3 women 4 men and 4 women 5 men and 5 women

Step 2: Calculate Combinations for Each Group Size

We will use the combination formula Cn k n! / (k!(n-k)!) to calculate the number of ways to choose k men from 8 and k women from 5 for each group size.

Groups with 1 man and 1 woman (k 1)

Number of ways to choose 1 man from 8: C8 1 8 Number of ways to choose 1 woman from 5: C5 1 5 Total for k 1: 8 * 5 40

Groups with 2 men and 2 women (k 2)

Number of ways to choose 2 men from 8: C8 2 8! / (2!(8-2)!) 28 Number of ways to choose 2 women from 5: C5 2 5! / (2!(5-2)!) 10 Total for k 2: 28 * 10 280

Groups with 3 men and 3 women (k 3)

Number of ways to choose 3 men from 8: C8 3 8! / (3!(8-3)!) 56 Number of ways to choose 3 women from 5: C5 3 5! / (3!(5-3)!) 10 Total for k 3: 56 * 10 560

Groups with 4 men and 4 women (k 4)

Number of ways to choose 4 men from 8: C8 4 8! / (4!(8-4)!) 70 Number of ways to choose 4 women from 5: C5 4 5! / (4!(5-4)!) 5 Total for k 4: 70 * 5 350

Groups with 5 men and 5 women (k 5)

Number of ways to choose 5 men from 8: C8 5 8! / (5!(8-5)!) 56 Number of ways to choose 5 women from 5: C5 5 1 Total for k 5: 56 * 1 56

Step 3: Sum the Total Combinations

Now we sum the total combinations for all group sizes: 40 280 560 350 56 1286

Conclusion

The total number of groups that can be formed from 8 men and 5 women with at least one member and an equal number of men and women is 1286.