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.
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; 129 is not the same as 291, 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, 333.
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 rpermutations of n or partial permutations, denoted as _{n}P_{r}, ^{n}P_{r}, 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 333. 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:
_{n}P_{r} = 

Or in this case specifically:
_{11}P_{2} = 

= 

= 11 × 10 = 110 
Again, the calculator provided does not calculate permutations with replacement, but for the curious, the equation is provided below:
_{n}P_{r} = n^{r}
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 _{n}C_{r}, ^{n}C_{r}, C_{(n,r)}, or C(n,r), or most commonly as simply
(  n  ) 
r 
_{n}C_{r} = 

Or in this case specifically:
_{11}C_{2} = 

= 

= 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:
_{n}C_{r} = 

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:
 ABC
 ACB
 BAC
 BCA
 CAB
 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! * (32)!) = 3.
The three possible combinations for selecting 2 elements from the given set of letters are:
 AB
 AC
 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.