Question 1023146
 
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.