SOLUTION: Suppose we draw a four card hand from a standard 52 card deck. A) how many different hands contain 3 cards of the same value? B) how many different hands contain 4 cards of th

Question 1023146: Suppose we draw a four card hand from a standard 52 card deck.
A) how many different hands contain 3 cards of the same value?
B) how many different hands contain 4 cards of the same value?
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

Question:
Suppose we draw a four card hand from a standard 52 card deck.
A) how many different hands contain 3 cards of the same value?
B) how many different hands contain 4 cards of the same value?

Solution:
Four-card hands means that order does not count.
There are 13 "values" in a deck, each in 4 different suits.
In the following, the combination "n choose r" is represented by
C(n,r)=n!/(r!(n-r)!)

A) 3 cards of the same value
For each "value", i.e. Ace, 2, 3....10,J,Q,K, there are C(4,3)=4 ways to choose the three cards, AND (13-1)=12 different ways to choose the fourth card.
There are thus 4*12=48 different hands that contain 3 card of the same value for each value. Since there are 13 "values" per pack, so there are 13*48=624 such hands.

B) 4 cards of the same value.
For each value, there is only C(4,4)=1 way to choose all four cards of the same value. Multiplied by 13 values, there are 13 such hands.
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