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Charley Crissman
Math 55 Discussion Notes
October 13th, 2010
Permutations and Combinations
Question:
Recall that a poker hand consists of 5 cards from a standard 52-card deck. How many ways are there to get a:
1. four-of-a-kind?
2. two-of-a-kind?
3. hand with no pairs?
4. ﬂush? (All cards the same suit)
5. straight? (A straight is ﬁve cards with consecutive ranks, but can be of any suit)
6. straight ﬂush? (combine the above two)
Four-of-a-Kind:
To get a four-of-a-kind, we need all four cards of a single denomination, plus one additional card. There are 13
denominations, and once we have chosen our denomination, there are 48 remaining cards not of that denomination
which we can take as our additional card. Hence our solution is:
13
·
48 = 624
Two-of-a-Kind:
There are several ways to compute this. Here is one method:
Choose a denomination for our pair (13 possible choices), then choose two cards of that denomination (
(
4
2
)
= 6
choices). Now, to choose the remaining three cards, we can choose their denominations (

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