6. If there are 12 teams in a basketball tournament and each team must play every other
team in the eliminations, how many elimination games will there be?
7. If there are 12 representatives in SSG (Supreme Student Government), how many
different sets of officers with 6 members can be formed.
8. A committee of 5 people is to be chosen from a group of 6 men and 4 women. How
many committees are possible if:
a) There are to be 3 men and 2 women?
b) There are two men only?
c) There is to be a majority of women?

Basketball is a game played between 2 teams. The problem is asking how many total games will there be. This can be seen as taking 2 teams to play from 12 possible teams. We don't care about the order, but we care about one team playing another team.

A combination is a selection of items from a collection such that the order of the selection does not matter. The formula for taking r things from n possible ones with no particular order is expressed by:

The "!" symbol is called "factorial". In simple terms,

For example, 4! is 4*3*2 = 24. Since it is a treated product, we can cancel out fractions with both x! on the numerator and denominator.

There are 2 ways to solve this problem.

This first way is via combination. We have n = 12, r = 2.

Substituting n = 12, r = 2 to the combination formula gives us:

There are 66 possible games in the 12-team tournament.

The second way is a bit intuitive. The first team, team 1, will play 11 of the teams, from team 2 to team 12. The second team, having played team 1 already, only plays 10 of the teams, from team 3 to team 12. This goes on until Team 11 plays 1 team, Team 12.

We can add the following games.

We got the same answer from earlier, 66.

7.) 2 sets with 6 members

8.) a.) 2 committees

b.) 3 committees

c.) 1 committee