When different orderings of the same items are to be counted individually, i.e. each listing represents a different scenario (mn is not the same as nm), the problem involves permutations.

When different orderings are not to be counted separately, i.e. the outcome, mn is equivalent to the outcome nm, the problem involves combinations.

Combination Formula: Different orders of the same items are not counted.  The combination formula is equivalent to dividing the corresponding number of permutations by r!. n: number of available items or choices r: the number of items to be selected     Sometimes this formula is written: C(n,r).

Taking the letters a, b, and c taken two at a time, there are six permutations: {ab, ac, ba, bc, ca, cb}.  If the order of the arrangement is not important, how many of these outcomes are equivalent, i.e. how many combinations are there?

ab = ba; ac = ca; and bc = cb
The three duplicate permutations would not be counted, therefore three combinations exist.

Calculate the value of ₇C_4.   Answer

Calculate the value of ₉C_5.   Answer

Determine whether the following scenarios represent permutations or combinations. Correct answers are provided at the end of this section.

1) Selecting two types of yogurt from the grocery's dairy case from a selection of nine.

2) Selecting your favorite yogurt and then your second favorite yogurt from a selection of nine.

3) Selecting three members from your class to work specific homework problems on the board.

4) Choosing two books to take with you on vacation from the nine books on your shelf.

5) Choosing three CDs to purchase from the music store.

6) Arranging seven photographs on a page of your senior memory book.

Answers

In how many ways can three class representatives be chosen from a group of twelve students?  If the order of the arrangement is not important, how many outcomes will there be?  Answer

Test your understanding of Combinations.
Practice Problems