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Permutations Vs Combinations

Difference between Permutation and Combination

In combination we are concerned only with the number of things n each selection whereas in permutation we also consider the order of things which makes each arrangement different The symbol ⁿP_r, denotes the number of permutations of n different things taken r at a time, whereas ⁿC_r denotes the number of combinations of n different things taken r at a time.

Enquiry: How can we find the number of ways of selection n things taken r at a time, when their order also matters?

This can be done by finding the number of permutations of n different things taken r at a time (n > r) or to find the value of ⁿP_r.

This is same as finding the number of ways in which we can fill up r places when we have n different things at our disposal. The first place may be filled up in n-ways, because we may take any one of the n things. When it has been filled up in any one of these ways, the second place can then be filled up in (n - 1) ways, and since each way of filling up the first place can be associated with each way of filling up the second, the number of ways in which the first two places can be filled up is given by the product n(n - 1) ways, And when the first two places have been filled up in any way, the third place can be filled up in (n - 2) ways. And reasoning as before, the number of ways in which three places can be filled up is n(n - 1)(n - 2).

Proceeding thus, and noticing that a new factor is introduced with each new place filled up, and that at any stage the number of factors is the same as the number of places filled up, we shall have the number of ways in which r places can be filled up, we shall have the number of ways in which r places can be filled up equal to n(n - 1)(n - 2) ......... to r factor, and the rth factor is n - (r - 1), or n - r + 1. Therefore the number of permutations of n things taken r at a time is

ⁿP_r = n(n - 1)(n - 2)......(n - r + 1).

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