Cardmeister: Combinations (2)

How many different ways can a person draw 5 cards from a standard deck of card without replacing them?

There are 52 * 51 * 50 * 49 * 48 different ways to select 5 cards (without replacement.)

The number 52 * 51 * 50 * 49 * 48 may also be represented as ^52! / _47! using factorial notation:

^52! / _47!	=	52 * 51 * 50 * 49 * 48 * 47!	= 52 * 51 * 50 * 49 * 48
		47!

Now we ask a different question. How many different sets of 5 cards may be chosen from a deck of 52 cards?

One must observe that
is the same hand as

since they are the same cards, just rearranged in a different order.

In Permutations, we learned that 5 cards may be arranged 5! different ways. Thus, we must divide the number of ways to select 5 cards from a deck of 52 by 5! to get the number of five card hands that are possible:

^52! / _47!	52!	the number of 5 card hands
______	= ______ =
5!	47! * 5!

When one is selecting a set of items from a larger group (i.e. cards from a deck) and the order of the items does not matter, it is a combination.

Janice is holding the 7 cards below in her hand, and tells you to pick three of them. How many different ways can you do this?

4 of Clubs

There are 7 ways to select the first card, 6 ways to select the second card, and 5 ways to select the third card. Therefore there are 7 * 6 * 5 = ^7! / _4! ways to select three cards. However, some of these selections will be the same! For example, the cards below show all the different ways to choose a three of clubs, a king of spades, and an eight of diamonds:

Since this one hand has 3! different permutations, we must divide ^7! / _4! by 3! to get the number of possible hands that can be selected from Janice.

^7! / _4!	7!	35 different hands may be picked from Janice
______	= ______
3!	4! * 3!

If one has n distinct objects, each selection or combination of r of these objects where order does NOT matter corresponds to

C( n, r ) =	^n! / _{(n - r)!}	n!
	______	= ______ =
	n!	r! * (n - r)!

Some things to note about the combinations formula:

C(n,r) is read " n choose r."

The symbol

( n )

r

is frequently used in place of C(n,r)
C(n,0) = 1. This makes sense, because if you have n objects, and you select zero of them, there is only 1 way to do it - don't select any of the objects!
Similarly, C(n, n) = 1. If you have n objects, and you select all n of them, there is only one way to do it - grab all of them!

Want to see if you have conquered combinations? Try out some Combinations Practice Problems.

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