Express how many ways you can select a representative Assume that a school has these three teams: Chess team with 10 members, Checkers team with 15 members, and College bowl team with 20 members.
In how many ways can we select representatives of the school if there should be:


*

*exactly 1 representative from each team.

*2 persons, and they should be from different teams.

*1, 2, or 3 persons but no two from the same team.

*2 persons, both from the same team.

*2 persons from any team.


Answers


*

*$20*15*10$ using the product rule

*$10 \cdot 15 + 10 \cdot 20 + 15 \cdot 20$

 A: Hints:
These will rely on various applications of the multiplication principle, permutations, combinations, and the summation principle.
1: There will be one representative from each team.  Who is the representative from the checkers team?  Who is the representative from the chess team?  Who is the representative from the College bowl team?
2: Consider the set of all 2 person groups with no restriction on what team they come from (same answer as problem 5).  How many of those are "bad" selections such that they came from the same team (same as problem 4)?  Use inclusion-exclusion to find the number of "good" selections.
3: Split this into cases.  If 1 person, choose that one person.  If 2 people, with no two from same team, this is same answer as problem 2.  If 3 people, this is one from each team (same as problem 1).
4: Split this into cases.  If both are from the chess team, which two are they?  If both are from the checkers team, which two are they?  If both are from the college bowl team, which two are they?  Use combinations to answer those questions, and finish by using summation rule.
5: There are no restrictions whatsoever, so 2 people out of the total population.
