EXAMPLE
1.5.18
1. Harpo, Groucho, Chico, Zeppo, Gummo,
Karl, and Skid have won three tickets to the opera. They will randomly choose
three people from their group to attend the opera. How many outcomes are
possible?
2. Suppose that instead of choosing 3
people to attend the opera, they decide instead to choose 4 people to not attend. How many outcomes are possible?
1. There are 7 people from whom to choose,
and we are choosing three of them.
Because all three people are receiving the same reward (they get to go
to the opera), the order in which they are chosen or listed is not important. (For instance, if Harpo, Groucho and
Chico go to the opera itÍs the same as if Chico, Harpo and Groucho go to the
opera.) Thus, this is a
combination problem.
C(7,3) = 35
2. Following the approach in #1 above, we should see that answer will be C(7,4). Another way to get the answer is to understand that the answer to this problem should be exactly the same as the answer to problem #1. Why? Because every time we select a three person group to go to the opera, we are automatically choosing a four-person group to not go to the opera. That is, for each of the 35 different 3-person groups in the answer to #1, there is a corresponding 4-person group in the answer to #2. Thus there must be exactly 35 4-person groups. Indeed, C(7,4) = 35.