Combinations and Permutations Calculator

Find the number of combinations and/or permutations that result when you choose r elements from a set of n elements.

For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

Choose the goal of your analysis (i.e., to compute combinations or permutations). Enter a value in each of the unshaded text boxes. Click the Calculate button to display the result of your analysis. Choose goal Count combinations Count permutations Number of sample points in set ( n ) Number of sample points in each combination ( r ) Number of combinations (n things taken r at a time)

Instructions: To find the answer to a frequently-asked question, simply click on the question. If none of the questions addresses your need, refer to Stat Trek's tutorial on the rules of counting or visit the Statistics Glossary. On-line help is just a mouse click away.

• What is a permutation?
• What is a combination?
• What is the difference between a combination and a permutation?

See also:

Rules of Counting | Statistics Glossary

What is a permutation?

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.

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

What is a combination?

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.

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

What is the difference between a combination and a permutation?

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.

1. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, and 7, if each digit can be used only once?
   
   Solution:
   
   The solution to this problem involves counting the number of permutations of 7 distinct objects, taken 3 at a time. The number of permutations of n distinct objects, taken r at a time is
   
   
   _nP_r = n! / (n - r)!
   ₇P₃ = 7! / (7 - 3)! = 7! / 4! = (7)(6)(5) = 210
   
   Thus, 210 different 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, and 7. To solve this problem using the Combination and Permutation Calculator, do the following:
   
   
   • Choose "Count permutations" as the analytical goal.
   • Enter "7" for "Number of sample points in set ".
   • Enter "3" for "Number of sample points in each permutation".
   • Click the "Calculate" button.
   
   The answer, 210, is displayed in the "Number of permutations" textbox.
   
   
2. The Atlanta Braves are having a walk-on tryout camp for baseball players. Thirty players show up at camp, but the coaches can choose only four. How many ways can four players be chosen from the 30 that have shown up?
   
   Solution:
   
   The solution to this problem involves counting the number of combinations of 30 players, taken 4 at a time. The number of combinations of n distinct objects, taken r at a time is
   
   
   _nC_r = n! / r! (n - r)!
   ₃₀C₄ = 30! / 4!(30 - 4)! = 30! / 4! 26! = 27,405
   
   Thus, 27,405 different groupings of 4 players are possible. To solve this problem using the Combination and Permutation Calculator, do the following:
   
   
   • Choose "Count combinations" as the analytical goal.
   • Enter "30" for "Number of sample points in set ".
   • Enter "4" for "Number of sample points in each combination".
   • Click the "Calculate" button.
   
   The answer, 27,405, is displayed in the "Number of combinations" textbox.