Difference between Permutations and Combinations

Each of the arrangements that can be made out of a given set of things, by taking some or all of them at a time, are called Permutations. The order in which arrangements are taken is important in a Permutation.

Each of the groups or selections (in any order) that can be made out of a given set of things by taking some or all of them at a time are called combinations. The order in which selections are made is not important in a Combination.

Example: Two letters a and b together form one group (combination), but they can be arranged in two different ways as ab and ba and thus there are total of  two arrangements (permutations).

Again, if we take three letters a, b, and c, then the number of groups taking two letters at a time is three i.e., ab, bc, and ca.

But each group gives rise to two different arrangements, hence the total number of arrangements = 6 i.e., ab, ba, bc, cb, ca, and ac.

Further, if we take four letters a, b, c, and d, then the combinations which can be made by taking two letters at a time are six in numbers.

ab, ac, ad, bc, bd, cd

And the permutations which can be made by taking two letters at a time are twelve in number

ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dc

Before we proceed on to study permutations and combinations in detail, we shall introduce a notation n! read as n factorial, which is very helpful in the study and calculation of permutations and combinations.

Combination is a way of selecting items from a collection of items in combination we do not look at the order of selecting items, but our main attention is on the total number of selected items from a given set of items. For example- suppose that we have three numbers say, a, b, and c. Then in how many ways we can select two numbers is known as a combination.