# Basic Counting Principles

**Sum Rule Principle:** Assume some event E can occur in m ways and a second event F can occur in n ways, and suppose both events cannot occur simultaneously. Then E or F can occur in m + n ways.

In general, if there are n events and no two events occurs in same time then the event can occur in n_{1}+n_{2}……….n ways.

**Example:** If 8 male processor and 5 female processor teaching DMS then the student can choose professor in 8+5=13 ways.

**Product Rule Principle:** Suppose there is an event E which can occur in m ways and, independent of this event, there is a second event F which can occur in n ways. Then combinations of E and F can occur in mn ways.

In general, if there are n events occurring independently then all events can occur in the order indicated as n_{1} x n_{2} x n_{3}………n ways.

**Example:** In class, there are 4 boys and 10 girls if a boy and a girl have to be chosen for the class monitor, the students can choose class monitor in 4 x 10 = 40 ways.

## Mathematical Functions:

**Factorial Function:** The product of the first n natural number is called factorial n. It is denoted by n!, read “n Factorial.”

The Factorial n can also be written as

**Example1:** Find the value of 5!

**Solution:**

5! = 5 x (5-1) (5-2) (5-3) (5-4) = 5 x 4 x 3 x 2 x 1 = 120

**Example2:** Find the value of

**Solution:** == 10 x 9=90

**Binomial Coefficients:** Binomial Coefficient is represented by n_{Cr} where r and n are positive integer with r ≤ n is defined as follows:

**Example:** 8_{C2} === 28.