# Permutation and Combination Calculator Permutation and Combination Calculator - Easily calculate permutations and combinations for any set of elements. Find the number of possible arrangements and selections. Fast, accurate, and efficient tool.

 Total Amount in a Set (n) Amount in each Sub-Set (r)

Related

Permutations and combinations are part of a branch of mathematics called combinatorics, which involves studying finite, discrete structures. Permutations are specific selections of elements within a set where the order in which the elements are arranged is important, while combinations involve the selection of elements without regard for order. A typical combination lock for example, should technically be called a permutation lock by mathematical standards, since the order of the numbers entered is important; 1-2-9 is not the same as 2-9-1, whereas for a combination, any order of those three numbers would suffice. There are different types of permutations and combinations, but the calculator above only considers the case without replacement, also referred to as without repetition. This means that for the example of the combination lock above, this calculator does not compute the case where the combination lock can have repeated values, for example, 3-3-3.

### Permutations

The calculator provided computes one of the most typical concepts of permutations where arrangements of a fixed number of elements r, are taken from a given set n. Essentially this can be referred to as r-permutations of n or partial permutations, denoted as nPr, nPr, P(n,r), or P(n,r) among others. In the case of permutations without replacement, all possible ways that elements in a set can be listed in a particular order are considered, but the number of choices reduces each time an element is chosen, rather than a case such as the "combination" lock, where a value can occur multiple times, such as 3-3-3. For example, in trying to determine the number of ways that a team captain and goalkeeper of a soccer team can be picked from a team consisting of 11 members, the team captain and the goalkeeper cannot be the same person, and once chosen, must be removed from the set. The letters A through K will represent the 11 different members of the team:

A B C D E F G H I J K   11 members; A is chosen as captain

B C D E F G H I J K   10 members; B is chosen as keeper

As can be seen, the first choice was for A to be captain out of the 11 initial members, but since A cannot be the team captain as well as the goalkeeper, A was removed from the set before the second choice of the goalkeeper B could be made. The total possibilities if every single member of the team's position were specified would be 11 × 10 × 9 × 8 × 7 × ... × 2 × 1, or 11 factorial, written as 11!. However, since only the team captain and goalkeeper being chosen was important in this case, only the first two choices, 11 × 10 = 110 are relevant. As such, the equation for calculating permutations removes the rest of the elements, 9 × 8 × 7 × ... × 2 × 1, or 9!. Thus, the generalized equation for a permutation can be written as:

nPr
 n! (n - r)!

Or in this case specifically:

11P2
 11! (11 - 2)!
=
 11! 9!
= 11 × 10 = 110

Again, the calculator provided does not calculate permutations with replacement, but for the curious, the equation is provided below:

nPr = nr

### Combinations

Combinations are related to permutations in that they are essentially permutations where all the redundancies are removed (as will be described below), since order in a combination is not important. Combinations, like permutations, are denoted in various ways, including nCr, nCr, C(n,r), or C(n,r), or most commonly as simply

 ( n ) r
. As with permutations, the calculator provided only considers the case of combinations without replacement, and the case of combinations with replacement will not be discussed. Using the example of a soccer team again, find the number of ways to choose 2 strikers from a team of 11. Unlike the case given in the permutation example, where the captain was chosen first, then the goalkeeper, the order in which the strikers are chosen does not matter, since they will both be strikers. Referring again to the soccer team as the letters A through K, it does not matter whether A and then B or B and then A are chosen to be strikers in those respective orders, only that they are chosen. The possible number of arrangements for all n people, is simply n!, as described in the permutations section. To determine the number of combinations, it is necessary to remove the redundancies from the total number of permutations (110 from the previous example in the permutations section) by dividing the redundancies, which in this case is 2!. Again, this is because order no longer matters, so the permutation equation needs to be reduced by the number of ways the players can be chosen, A then B or B then A, 2, or 2!. This yields the generalized equation for a combination as that for a permutation divided by the number of redundancies, and is typically known as the binomial coefficient:

nCr
 n! r! × (n - r)!

Or in this case specifically:

11C2
 11! 2! × (11 - 2)!
=
 11! 2! × 9!
= 55

It makes sense that there are fewer choices for a combination than a permutation, since the redundancies are being removed. Again for the curious, the equation for combinations with replacement is provided below:

nCr
 (r + n -1)! r! × (n - 1)!

### Permutation and Combination Calculator Example

Imagine you have a set of letters: A, B, C.

To calculate permutations, which refers to the number of possible arrangements where order matters, we can use the permutation formula. The number of permutations can be calculated by multiplying the total number of elements by one less than each subsequent element, and so on. In this case, since we have 3 elements, the number of permutations would be 3! = 3 x 2 x 1 = 6.

The six possible permutations for the given set of letters are:

1. ABC
2. ACB
3. BAC
4. BCA
5. CAB
6. CBA

To calculate combinations, which refers to the number of possible selections regardless of order, we can use the combination formula. The number of combinations can be calculated by dividing the number of permutations by the factorial of the selected elements. In this case, if we want to select 2 elements from the set of 3, the number of combinations would be 3C2 = 3! / (2! * (3-2)!) = 3.

The three possible combinations for selecting 2 elements from the given set of letters are:

1. AB
2. AC
3. BC

Therefore, using a permutation and combination calculator with the provided table, you can easily determine the number of permutations and combinations for any given set of elements.