**PERMUTATION:**

**Suppose an event can be chosen in***p*different ways. Another independent event can be chosen in*q*different ways. Then the two events can be chosen successively in*p.q*ways.

*Example Problem:*

On
a trip a man took 3 suits, 2 ties, and 2 hats. How many different
choices of these items of clothing are possible:

Answer:
There
are 3.2.2 = 12 different ways.

**The number of permutations of***n*objects, taken*r*at a time, is defined by:

**P(n,r) = n! / (n-r)!**

*Example Problem:*

How
many different three-letter letter patterns can be formed using the
letters a,b,c,d, and e without repetition?

Answer:
Using P(n,r)
= n! / (n-r)! formula,
P (5,3) = 5! / (5-3)! = (5.4.3.2.1) / (5-3)! = 60.

*Meaning there are 60 ways to arrange the letters a,b,c,d and e three at a time.*

**The number of permutations of***n*objects, taken*n*at a time, is defined by:

**P (n,n) = n!**

*Example Problem:*

How many different sample of size 4 can we form the wooden block with letters A, B ,C, D?

Answer: Using P (n,n) = n! formula, P(4,4)= 4! / (4-4)! = 24

Answer: 5! / 2!2! = 30

**PERMUTATION WITH REPETITIONS AND CIRCULAR PERMUTATIONS**

- The number of Permutations of n objects of which p are alike and q are alike is found by:

**n! / p!q!**

*Example Problem:***How many Five-Letter patterns can be formed from the letters of the word teeth?**

Answer: 5! / 2!2! = 30

- Suppose n objects are arranged in a circle.Then There are n! / n or (n-1)! permutations of the n objects around the circle:

*Example Problem:*

A vending machine has six different items on a revolving tray. How many ways can the items be arranged on a tray?

Answer: (6-1)! = 120