GMAT Exam: A Simple Guide To Arithmetic - Counting Techniques

The product rule for counting: For a sequence of k tasks, if a first task can be done in any one of n1 different ways, and for each of these ways, a subsequent second task can be done in any one of n2 different ways, and for each of these ways, a subsequent third task can be done in any one of n3 different ways, and so on to the kth task, which can be done in any one of nk different ways, then the total number of different ways the sequence of k tasks can be done is n1 × n2 × n3 × ··· × nk. Here are examples:

- The number of 3-digit codes that are possible using the digits 1 through 5, if repetitions of digits are allowed, is 5 × 5 × 5 = 125 (because you have 5 ways to pick each of the 3 digits).
- not allowed, is 5 × 4 × 3 = 60 (because once a particular digit is selected, it is no longer available to be picked).
- cat, if repetitions of letters is not allowed, is 3 × 2 × 1 = 6.

A permutation is an ordered arrangement of distinct objects in which repetition of objects is not allowed and different orderings of the same objects are counted as different outcomes.

For instance, cat, cta, act, atc, tca, and tac are the six permutations of the letters in the word cat.

For instance, cat and act are different permutations.

Through a direct application of the product rule for counting, the number of permutations of n distinct objects is n!, where n! = (n)(n – 1)(n – 2) ··· (3)(2)(1).

Read n! as “n factorial.”

A factorial is the product of all positive integers less than or equal to a given positive integer.

Exception: By definition 0! = 1. For example, the number of different ways to arrange the three letters in the word cat, if repetition of letters is not allowed, is 3!, which is 3 × 2 × 1 = 6.

A combination is an arrangement of distinct objects in which repetition of objects is not allowed and different orderings of the same items are considered to be the same arrangement.

That is, when the order in which you make a selection for an arrangement of objects does not determine different outcomes, the arrangement is a combination of the objects.

There is only one combination of n distinct objects because all the different ways you can arrange the n objects are not counted as different combinations.

For example, there is only one combination of the three letters in the word cat.

The six arrangements, cat, cta, act, atc, tca, and tac, are considered to be the same.

When you select r objects from n distinct objects without repetition, the number of combinations is nCr, where .

Tip: The notation nCr also is written as .

For example, the number of ways to put five people in pairs is


For combinations involving relatively small numbers like those given in this example, you might prefer to figure out the answer by listing the different combinations.

Designate the people as A, B, C, D, and E.

Then systematically list all of the 10 ways to match them two at a time: AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.

Be careful when listing the combinations.

You might overcount or undercount.

Remember, AB and BA, AC and CA, and so forth are not different combinations.

In general, using the combination formula saves time and is an accurate way to obtain the answer.