GMAT Exam: A Simple Guide To Solving Word Problems - Permutations and Combinations

Permutations
A permutation is a selection in which order is important. That is, different orderings of the same elements are counted separately.

or example, two permutations of the numbers from one to five are 12345 and 53142.

On the GMAT, use the product rule for counting to work permutation problems.

Make sure that:
- mutually different items (that is, no two are alike).
- without replacement.
- separately.

See the guide to “Counting Techniques” for an explanation of the product rule for counting.

Some situations that indicate you might have a permutation problem are the following:

creating codes, passwords, or license plates; making words; assigning roles; filling positions; making ordered arrangements of things (people, objects, colors, and so on), selecting persons or things as first, second, third, and so on; distributing items among several objects or people; and similar scenarios.

For example, the number of different ways a club of 20 members can select a president, vice-president, and secretary from its membership if no person holds more than one office and all members are eligible for any one of the three positions is 20 × 19 × 18 = 6,840.

(There are 20 members from which to select a president. After that position is filled, there are 19 members from which to select a vice-president. After the first two positions are filled, there are 18 members from which to select a secretary.)

Combinations
A combination is a selection in which order is not important.

For example, the set of vowels a, e, i, o, and u is the same as the set of vowels i, a, u, o, and e.

However, two different combination consisting of four vowels are the set a, e, i, and o and the set a, e, i, and u.

On the GMAT, to work combination problems use the combination formula


Make sure that:
- mutually different items (that is, no two are alike).
- without replacement.
- Different orderings of the same items are not distinguished as being different from each other.

See “Counting Techniques” for an explanation of the combination formula.

Some situations that indicate you might have a combination problem are the following:

making a collection of things (books, coins, and so on); selecting a committee; choosing questions from a test; counting the number of subsets of a given size from a set; dealing hands from a deck of cards; selecting pizza toppings; listing the combinations from a set of items; choosing students for groups; and similar scenarios.


Example: The number of different ways a club of 20 members can select a 3-member officer-nominating committee from its membership if all members are eligible to serve on the committee is

See “Counting Techniques”guide for additional discussion of permutations and combinations.