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 _{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
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:
_{n}P_{r} =
n!
(n - r)!
Or in this case specifically:
_{11}P_{2} =
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:
_{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
. 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:
_{n}C_{r} =
n!
r! × (n - r)!
Or in this case specifically:
_{11}C_{2} =
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: