Answer the following counting/probability questions. Give reasons for your answe
ID: 2892204 • Letter: A
Question
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 exactly two cards of the same rank (e.g., two 8s) and three of a kind means exactly 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].
Explanation / Answer
A full house has the form of one pair plus a three-of-a-kind then
there are 13 * 12 = 156 choices for the ranks of the pair
and
the three-of-a-kind (note that we don’t need to
remove permutations from the choices because there is a difference in which of the pair
or three-of-a-kind gets which rank. For example a full house consisting of two 2s and three 8s is
different than one consisting of two 8s and three 2s).
There are 4C2 = 6 choices for the
pair in its rank and
4C3 = 4 choices for the three-of-a-kind .
Therefore there are 12 * 13 * 4C2*
4C3 = 3744 possible full houses.