Question

There is a group A of 5 boys and 3 girls and another group B of 5 boys and 6 girls. How many ways can we invite 4 boys and 4 girls for a party with 5 from group A and 3 from group B?

Hint:

When selecting items from different groups, break down the problem into parts, solving each for one group at a time, and then multiply the results.

Answer 1

We need to select 4 boys and 4 girls for the party. According to the problem, we must select 5 boys from group A and 3 boys from group B. - First, we select 4 boys from group A, which consists of 5 boys. The number of ways to do this is: \[ \binom{5}{4} \] - Then, we select 3 boys from group B, which consists of 5 boys. The number of ways to do this is: \[ \binom{5}{3} \] For the selection of girls, we need to choose 4 girls. Since there are 3 girls in group A and 6 girls in group B: - We need to select 3 girls from group B, which consists of 6 girls. The number of ways to do this is: \[ \binom{6}{3} \] Thus, the total number of ways to invite the boys and girls is the product of the combinations: \[ \binom{5}{4} \binom{5}{3} \] Thus, the answer is \( \binom{5}{4} \binom{5}{3} \).