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Suppose we are selecting a few items from a large number of distinct items, and we want to know how many different ways this can be done. We are trying to find the number of combinations.
Combinations give the number of ways r items can be selected from n items, when the order of the items does not matter.
Suppose you want to buy 2 cars from a set of 4 distinct choices: {Honda, Toyota, Mercedes, BMW}. How many ways can you select two cars from this set?
Here are all the different ways to do this:
Each of the possible selections in this list is called a combination. Therefore, the number of combinations is 6.
The order does not matter. When dealing with combinations, the order in which the selections are made is not important.
Both of these arrangements represent 1 combination, so each pair is counted only once.
There are three ways to write the number of combinations.
All three are read as "the number of combinations of n items selected r at a time."
Let us revisit our example about selecting 2 cars from a set of 4 choices, and use the formula to find the number of combinations. In this case, n = 4 and r = 2.
4! = 4 × 3 × 2!, so the 2! on the top and the 2! on the bottom cancel.
(42)= 4 × 3 × 2!2!(2)! = 4 × 32!The formula gives 6 combinations, the same as the 6 pairs we listed in the car example.
Learn what n! means, so you can use the combinations formula.
Find the number of combinations for any n and r.