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 FrequentlyAsked 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.




Instructions: To find the answer to a frequentlyasked
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. Online help is just a mouse click away.
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 2letter 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.
 How many 3digit 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
_{n}P_{r} = n! / (n  r)!
_{7}P_{3} = 7! / (7  3)! = 7! / 4! = (7)(6)(5) = 210
Thus, 210 different 3digit 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.

The Atlanta Braves are having a walkon 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
_{n}C_{r} = n! / r! (n  r)!
_{30}C_{4} = 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.