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Meaning of Permutation and Combination

If we go by the dictionary meaning of the words permutation and combination, then permutation is the number of ways in which a set or a number of things can be put in an order or arranged and; combination refers to the number of ways in which a group of things can be chosen from a larger group without regard to their arrangement.

Let us go through an example. Suppose there are four different batsmen A, B, C and D and we have to select a group of three batsmen out of these four. Now, we can select any combination of three batsmen so that no set of batsmen has all the same three batsmen. These set of batsmen will be — ABC, BCD, ABD, ACD. This is a case of combination as for every set of selection of three batsmen, order of selection does not play any role (i.e., we can select anybody—first or second or third—and it does not create any difference in the final selection as well as in the total number of selections).

Now, if we try to define their batting order also, i.e., who bats first and second and so on, then corresponding to every selection of a set of three batsmen, we will have six different arrangements of their batting order. It can be seen below that corresponding to the selection of ABC as a team, following is the list of different batting orders:
This is a case of permutation since the order of occurrence has become important. Since there are four different ways of selecting a group of three batsmen and every selection can be arranged in 6 different ways, so the total number of ways of arranging 3 batsmen (or, distinct things) out of 4 batsmen (or, distinct things) = 4 × 6 = 24 ways.
Permutation and combination can be better understood through the examples of hand-shake and gifts exchange also. Assume that there are 20 persons in a party and everybody shakes hand with each other and also presents a gift. Now if we take a case of two persons A and B, then the event of shaking hand between them is a case of combination because when A shakes hand with B or B shakes hand with A, the number of hand shake is just one. So, there is no order as such and hence it is a case of combination.
Similarly, the event of presenting the gift is a case of permutation because the gift given to B by A and the gift given to A by B are two different gifts. So, the order of case plays a role here and hence it is a case of permutation.

n! = Product of all the natural numbers from n to 1 = n (n−1) (n−2) (n−3)…× 3 × 2 × 1.
0! = 1
Factorials are defined only for whole numbers, and not for negative numbers or fractions (≠ whole numbers).

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