FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.392.6 PERMUTATION AND COMBINATION
2.6.1. PERMUTATION :
Definition :
The different arrangements which can be made out of a given set of things, by taking some or all of them
at a time are called permutations.
Thus the permutations of three letters a, b, c taking one, two or three at a time are respectively :
one : a b c
two : ab bc ca ba cb ac
three : abc bca cab acb bac cba
The number of permutations of n different things, taken r at a time, usually symbolised by nPr or nPr.
Thus the number of arrangements (or Permutations) of 3 things taken 1, 2 and 3 at a time are
respectively :^3 P 1 ,^3 P 2 and^3 P 3.
General Principle :
If one operation can be performed in m different ways and corresponding to any one of such operations if
a second operation can be performed in n different ways, then the total number of performing the two
operations is m × n.
The above principle is applied in the following theory of permutations.
Permutations of things all different :
To find the number of permutations of n different things taken r ( r ≤ n) at a time.
This is the same thing of finding out the number of different ways in which r places can be filled up by the n
things taking one in each place.
The first place can be filled up in n ways since any one of the n different things can be put in it.
When the first place has been filled up in any of these n ways, the second place can be filled up in
(n – 1) ways, since any one of the remaining (n – 1) things can be put in it.
Now corresponding to each of filling up the first place, there are (n – 1) ways of filling up the second, the
first two places can be filled up in n (n – 1) ways.
Again, when the first two places are filled up in any one of the n (n – 1) ways, the third place can be filled
up by (n – 2) ways, for there are now (n – 2) things, at our disposal, to fill up the third place, Now corresponding
to the each way of filling up the first two places, there are clearly (n – 2) ways of filling up the third place.
Hence the first three places can be up in n (n –1) (n – 2) ways.
Proceeding similarly and noticing that the number of factors at any stage, is always equal to the number of
places to be filled up, we conclude that the total number of ways in which r places can be filled up.
= n (n – 1) (n – 2) ........ to r factors
= n (n – 1) (n – 2) ......... {n – (r – 1)}
= n (n – 1) (n – 2) ........ (n – r + 1)
Hence, using the symbol of permutation, we get,
nPr = n (n – 1) (n – 2) ... (n – r + 1)
Cor. The number of permutation of n different things taking all at a time is given by