Permutation and Combination Formula

In our day to day life, we are facing many situations in which we have to make the arrangements of some objects or selection of some objects taken from a collection. For example, selecting 3 balls from a set of 10 balls or arranging 5 numbers in all possible orders. These can be computed with the help of permutation and combination. In mathematics as well as in statistics these two computations are very useful for many applications. In this article, we will see the concepts of permutation and combinations as well as permutation and combination formula with example. Let us start learning the concept.

Permutation and Combination Formula

What are permutations and combinations?

Permutations and combinations are the various ways in which objects from a given set may be selected. Normally it is done without replacement, to form the subsets. This selection of subsets is known as permutation when the order of selection is important, and as combination when order is not an important factor.

Thus Permutation is the different arrangements of a given number of elements taken some or all at a time. For example, if we have two letters A and B, then there are two possible arrangements, AB and BA. On the other hand, the combination is the different selections of a given number of objects taken some or all at a time. For example, if we have two alphabets A and B, then there is only one way to select two items, we select both of them.

The difference with Example:

The conceptual differences between permutations and combinations can be illustrated by having all the different ways in which a pair of objects can be selected from five distinguishable objects as A, B, C, D, and E. If two letters were selected and the order of selection are important then the following 20 outcomes are possible as AB, BA, AC, CA,  AD,  DA,  AE,  EA, BC, CB, BD, DB, BE, EB, CD, DC, CE, EC, DE, ED.

For combinations, k elements are selected from a set of n objects to produce subsets without bothering about ordering. In contrast with the previous permutation example with the corresponding combination, the AB and BA will be no longer distinct selections. Thus by eliminating such cases there remain only 10 different possible groups, which are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.

Permutation and Combination Formula:

The formulas for nPk and nCk are popularly known as counting formulae. This is because these can be used to count the number of possible permutations or combinations in a given situation.

In general, if there are n objects available. And out of these to select k, the number of different permutations possible is denoted by the symbol nPk.

A formula for its computation is

nPk = \(\frac{n!}{(n-k)!}\)

Also, the number of subsets, denoted by nCk, and read as “n choose k.” will give the combinations. It is obvious that this number of subsets has to be divided by k! , as k! arrangements will be there for each choice of k objects. Thus,

nCk = \(\frac{n!}{(n-k)!* k!}\)

Hence, if the order doesn’t matter then we have a combination, and if the order does matter then we have a permutation. Also, we can say that a permutation is an ordered combination.

Solved Examples

Q. In a lucky draw of ten names are out in a box out of which three are to be taken out. Find the number of total ways in which three names can be taken out.

Solution: Here three names will be taken out. Thus selection is there without having botheration about ordering the selection.

Thus, the possible number of ways for finding three names out of ten from the box will be :

C (10, 3)

= \(\frac{10!}{(10-3)! * 3!}\)

= \(\frac{10 * 9 * 8}{(3*2*1)}\)

=  120