# Concept of Permutations

The word permutation means **arrangement**. For example, given 3 letters a, b, c suppose you arrange them taking 2 at a time. The various arrangements are: ab, ba, bc, cb, ac, ca.

Hence the number of arrangements of 3 things taken 2 at a time is 6. This is written as

^{3}P_{2} = 6

**Definition**

The number of arrangements that can be made out of n things taking r at a time is called the number of permutations of n things taken r at a time.

**Notation**

If n and r are positive integers such that 1 ≤ r ≤ n, then the number of all permutations of n distinct things, taken r at a time is denoted by the symbol P(n,r) or ^{n}P_{r}.

In permutations the order of arrangement is taken into account; when the order is changed, a different permutation is obtained.

### Properties of Permutations

**1.** Let r and n be positive integers such that 1 ≤ r ≤ n. Then, the number of all permutations of n distinct things taken r at a time is given by ^{n}P_{r}.

**2.** Let r and n be positive integers such that 1 ≤ r ≤ n. Then

^{ n}P_{r} = n!/(n − r)!

**3.** The number of all permutations of n distinct things, taken all at a time is n!.