Week One Solutions
1)
Consider a group of five men and four women with the following hand shaking rules and
calculate how many possible handshakes there are:
•
Only men shake hands with each other. How many possible handshakes are there?
Handshakes are between all possible pairs of men so the number of handshakes is:
•
Men shake hands with both men and women but women do not shake hands with other
women. How many possible handshakes are there?
There are 10 handshakes between men. The “mixed sex” handshakes (i.e. those between one
man and one woman) number
All together there are 10+20=30 handshakes.
•
Everyone shakes hands with everyone else. How many possible handshakes are there?
This is the number of pairs of nine people or
•
Men shake hands with men and women with women but not men with women.
There are handshakes between men and handshakes between women. The total number is 16.
2)
Seven women are to play in a tennis tournament. Figure out the number of different possible
ways to start the first round of a tournament (three pairs) under the following rules:
•
It does not matter where the pairs play or who serves first.
First we choose who will play. There are choices. Then we consider the oldest player. She has five
choices who to play against. The oldest of the four remaining players has three possible opponents.
The last two players must play against each other – one possibility. Thus the number of ways to start
the tournament is
Another way to count the pairings is to choose who will play on the first court (choices), then the
pair that will play on the second court (choices) and finally the final pair (choice). But we need to
divide by the possible orders in which the same three pairs can occur. Thus the number of ways is
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 Spring '11
 FredPhelps
 Math, partner, Natural number

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