How many different 5 card hands include the ace and king of spades??

Question

How many different 5 card hands include the ace and king of spades??

Answer 1

You already have the ace and king of spades, so we need to find out how many three card combos without those cards there are:

52 cards in a deck - 2 cards = 50 cards left

50 (number of choices for first card) * 49 (number of choices for second card) * 48 (number of choices for third card) = 117600

Now we need to divide that by the amount of times a combination can come up (for example, 1,2,3; 1,3,2; and 2,1,3 are all the same)

3 (number of choices for first card) * 2 (blah blah you heard it before) * 1 = 6

117600 / 6 = 19600

so there are 19600 different card hands that include the ace and king of spades.

Answer 2

If you have n objects and chose r of them, the number of combinations is:

n! / ( r! (n-r)! )

this can be written as nCr

there are two cards, the ace and king of spades that will be in the hand, there are 50 C 3 other combination of cards.

50! / (3! (50-3)!)

= 50! / (3! (47)!)

= 50 * 49 * 48 / (6)

= 19600