Statistical Methods

Unit 1 - Basic concepts of probability

1.5.Counting Techniques - Permutations and Combinations
Some useful techniques for counting the number of events satisfying certain conditions is presented in this section. These techniques are useful in computing the probability of an event when the total number of possible events is large.

Factorials
For any given positive integer n, the product of all the whole numbers from n down through 1 is called n factorial and is written as n!

For example,
5! = 5x4x3x2x1=120
8! = 8x7x6x5x4x3x2x1 = 40320
And in general
n! = n(n-1)(n-2) .............2x1
By definition, 0! = 1
Further,
n! = n(n-1)!
= n(n-1)(n-2)! And so on

Factorials are useful in finding the number of ways objects can be arranged in a line. For example, suppose that there are 3 containers of culture media, each of which is inoculated with a different organism. These culture media can be placed in a line on a platform in 3! = 6 ways. If the 3 media are designated as a, b and c, then 6 arrangements are abc, bca, cab, cba, bac, acb.

Combinations
Defined

Number of ways of selecting ‘r’ things out of ‘n’ things

Permutations
Defined

Number of ways of arranging ‘r’ things out of ‘n’ things