## Counting Permutations

Permutations is that part of statistics involving arrangements of objects, some of which fall in the same group and some of which fall in different groups. Each object is distinct from all the other objects, so we can tell each one apart, and if two objects are interchanged, this is a different arrangement of the objects. There are a wide variety of questions that may be asked.

We may be asked how many arrangements can there be if objects in a group are arranged in a line. This is the simplest question – the answer is since the first in line can be chosen from candidates, the second from candidates, the third from candidates etc. Continuing in this way we find there are possible arrangements altogether.

Suppose that instead of a single group there are two groups. The first group has objects and the second group has objects. If the two groups must be arranged separately, with the first group together and the second all together, then the first may be arranged in ways and the second in ways, so the total number of arrangements is If We can have the first group either first or second this introduces another factor of 2 so there are arrangements altogether.

In general if we have m groups of objects with objects respectively then the number of arrangements of all the objects is with the groups in the natural order, with the first group first, second groups second, third group third etc. If we allow the order of the groups to change – as opposed to the objects within the groups – this introduces another factor of since there are groups.

Hence the number of arrangements of k groups of objects, with objects in group 1, objects in group 2, objects in group 3, is If we treat all the objects as part of one single group then there are elements altogether, and there are arrangements.

Suppose now that we have n objects arranged in a circle. You might imagine there are n! Possible arrangements but you are wrong. We must divide by a factor n because the ring can be rotated so that abcde for example is the same as bcdea, cdeab, deabc, eabcd and by a factor 2 because the ring can be reflected. Hence there are arrangements altogether. If a reflection is the same as a rotation – this is a special case. There is only one arrangement. 