What is a permutation in math?

Answer 1

A permutation of items is a selecting of those items in order.

For example all the possible permutations of all the digits 123 are:

123, 132, 213, 231, 312, 321

For example, all the possible permutations of all the letters word are:

word, wodr, wrod, wrdo, wdor, wdro

owrd, owdr, orwd, ordw, odwr, odrw

rwod, rwdo, rowd, rodw, rdwo, rdow

dwor, dwro, dowr, dorw, drwo, drow

As you can see, the more items there are to permutate, the larger the number of permutations possible, and the greater the chance in making a mistake in listing them all out.

There is a quick way to see how many permutations there are possible without having to write them all out.

If you have n items, then any can be selected as the first item, giving n different selections for the first item.

For each of those n first items selected, there are now (n-1) items left and any of them can be selected as the second item, giving n x (n-1) ways of selecting the first two.

For each of those two selected items, there are now (n-2) items left and any of them can be selected as the third item, giving n x (n-1) x (n-2) ways of selecting the first three.

This can be continued until you have selected all the items you want.

Thus the number of was of selecting r items from n is:

n x (n-1) x (n-2) x ... x (n-r+2) x (n-r+1)

where each term is one less than the previous, with the last one the difference between n and r plus one. If you are selecting all the items with r=n, then this becomes:

n x (n-1) x (n-2) x ... x 2 x 1

which is known as n factorial and written as:

n!

with an exclamation mark following the number.

For the examples given, with the digits 123 there are 3 items, so they can all be selected, or permutated in 3 x 2 x 1 = 6 ways; similarly with the letters word there are 4 x 3 x 2 x 1 = 24 different permutations.