jmc

Question

My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team to have more girls than boys?

Accepted Answer

We do this problem with three cases.

Case 1: 4 girls, 2 boys on the team.

With 4 girls on the team, there are

(4 8) = 70

ways to pick the girls, and

(2 6) = 15

ways to pick the boys, for a total of

70 \times 15 = 1050

.

Case 2: 5 girls, 1 boy on the team. With 5 girls on the team, there are

(5 8) = 56

ways to pick the girls, and

(1 6) = 6

ways to pick the boy, for a total of

56 \times 6 = 336

.

Case 3: 6 girls on the team. With 6 girls on the team, there are

(6 8) = 28

ways to pick the girls on the team.

This gives us a sum of

1050 + 336 + 28 = 1414

.