Combinations and Permutations Calculator

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement.

For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can arrange 2 letters from that set. Each possible arrangement would be an example of a permutation. The complete list of possible permutations would be: AB, AC, BA, BC, CA, and CB.

When statisticians refer to permutations, they use a specific terminology. They describe permutations as n distinct objects taken r at a time. Translation: n refers to the number of objects from which the permutation is formed; and r refers to the number of objects used to form the permutation. Consider the example from the previous paragraph. The permutations were formed from 3 letters (A, B, and C), so n = 3; and each permutation consisted of 2 letters, so r = 2.

Note that AB and BA are considered to be two permutations, because the order in which objects are selected matters. This is the key distinction between a combination and a permutation. A combination focuses on the selection of objects without regard to the order in which they are selected. A permutation, in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged. Bottom line: AB and BA represent two permutations, but only one combination.

For an example that counts permutations, see Sample Problem 1.

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected.

For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC.

When statisticians refer to combinations, they use a specific terminology. They describe combinations as n distinct objects taken r at a time. Translation: n refers to the number of objects from which the combination is formed; and r refers to the number of objects used to form the combination. Consider the example from the previous paragraph. The combinations were formed from 3 letters (A, B, and C), so n = 3; and each combination consisted of 2 letters, so r = 2.

Note that AB and BA are considered to be one combination, because the order in which objects are selected does not matter. This is the key distinction between a combination and a permutation. A combination focuses on the selection of objects without regard to the order in which they are selected. A permutation, in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged. Bottom line: AB and BA represent two permutations, but only one combination.

For an example that counts the number of combinations, see Sample Problem 2.

A combination is a selection of all or part of a set of objects, without regard to the order in which they were selected. This means that XYZ is considered the same combination as ZYX.

The number of combinations of r objects that can be selected from a set of n objects is denoted by _nC_r. And the formula for computing that number is:

_nC_r = n(n - 1)(n - 2) ... (n - r + 1)/r! = n! / r!(n - r)!

Note: In the formula above, n! refers to n factorial, where n! is equal to n(n-1)(n-2) ... (3)(2)(1).

A permutation is a selection of all or part of a set of objects, with regard to the order in which they were selected. This means that XYZ is considered a different permutation than ZYX.

The number of permutations of r objects that can be selected from a set of n objects is denoted by _nP_r. And the formula for computing that number is:

_nP_r = n(n - 1)(n - 2) ... (n - r + 1) = n! / (n - r)!

Note: In the formula above, n! refers to n factorial, where n! is equal to n(n-1)(n-2) ... (3)(2)(1).

The distinction between a combination and a permutation has to do with the sequence or order in which objects appear. A combination focuses on the selection of objects without regard to the order in which they are selected. A permutation, in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged.

For example, consider the letters A and B. Using those letters, we can create two 2-letter permutations - AB and BA. Because order is important to a permutation, AB and BA are considered different permutations. However, AB and BA represent only one combination, because order is not important to a combination.

The Combinations and Permutations Calculator uses E notation to express very large numbers. E notation is a way to write numbers that are too large or too small to be concisely written in a decimal format.

With E notation, the letter E represents "times ten raised to the power of". Here is an example of a number written using E notation:

3.02E+12 = 3.02 * 10¹² = 3,020,000,000,000

When the number of combinations or permutations is displayed as an ordinary integer, the result is exact. When the number of combinations or permutations is displayed using E notation, the result is not exact; but it is a very good approximation, accurate to 16 decimals.