Answer the following counting/probability questions. Give reasons for your answers. You may use any of the Propositions we proved about permutations and combinations.
Combinatorics Problem Set
Answer the following counting/probability questions. Give reasons for your answers. You may
use any of the Propositions we proved about permutations and combinations. 1. In poker, a pair means two cards of the same rank (e.g., two 8s) and three of a kind
means three cards of the same rank (e.g., three kings). A full house is three of a kind
and a pair. How many ways are there to draw a full house? [Hint: count the number of
ways to draw three of a kind and then count the number of ways to draw a pair]. 2. A club has twelve members. The members must choose three officers?a president, a
vice president, and a treasurer. How many ways are there to choose officers? 3. To play a certain lottery, contestants choose 6 numbers between 1 and 100. What is the
probability of winning this lottery? 4. Two rival factions have been fighting a civil war. They each send two representatives to
discuss a peace settlement. The representatives are to be seated around a circular
table, so that each one sits next to both their ally and an enemy. How many seating
arrangements are there? [Note: the position of the representatives at the table does not
matter, only their positions relative to one another].
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