Permutations

What is a permutation ? Permuations come in various disguises. Suppose three friends A, B and C have to stand in line for a photograph. In how many order can they stand?

Some of the possible arrangements (from left to right) are A, B, C: A, C, B: B, A, C B, C, A: C, B, A: C, A, B. Thus there are six possible ways in which they can arrange themselves for the photograph. Thus if objects have to be arranged in a row there are × × = ! possible permutations.

The number of permutations of objects taken all at a time is × × × = ! Thus if n objects have to be arranged in a line there are n × ( n − ) × ( n − ) × · · · × × × = n ! possible arrangements or permutations. Suppose you have letters A,B,C,D,E,F and G.

We want to make a letter string. We have choices for the 1st letter. Having chosen the first letter, we have choices for the second letter. Proceeding this way, we have choices for the 4th letter.

. More generally, the number of distinct permutations of r objects which can be made from n distinct objects is n ! ( n − r )! .

It is denoted by n P r . The formal proof of this result will be proved in this section.