# Factorial Notation

In this section we introduce a very useful notation, called the factorial notation. This notation is very convenient for representing the product of first n natural numbers.

1, 2, 3, ...., n.

**Definition**

If n is a natural number, then n factorial, denoted by n! orLn is defined to be the product of 1 2 3 (n-1)n

That is n! = 1 2 3 (n-1) n

As a special case, we define zero factorial to be 0! = 1.

We now list values of n! for some values of n.0! = 1,

1! = 1,

2! = 1 2 = 2

3! = 1 2 3 = 6,

4! = 1 2 3 4 = 24

5! = 1 2 3 4 5 = 120,

6! = 1 2 3 4 5 6 = 720

We can also define n! recursively as follows:

**Recursive Definition of Factorial**

We define

0! = 1 and n! = n(n-1) ! for n â‰¥ 1

Thus 7! = 7 (6!) = 7 (720) = 5040,

and 8! = 8(7!) = 8 (5040) = 40320

**Example**

- Compute 4 ! + 3! Is 4! + 3! = 7!?
- Evaluate when (a) n = 5, r = 2, (b) n = 6, r = 3

**Solution**

- We have 4! + 3! = 24 + 6 =30.Also 7! = 7(6!) = 7 6 5! = 7 6 5 (4!) = 7 6 5(24) = 5040

Therefore, 4! + 3! â‰ 7! - (a) When n = 5, r = 2, then

=== = 20

(b) When n = 6, r = 3, then

=== = 120

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# Properties of Factorials

- It is a natural number.
- If n â‰¥ 5, then n! ends in a zero.
- If n > 1 then n! is divisible by 2, 3, 4, ... , n.
- If n â‰¥ 1, then n! is divisible by r ! for 1 â‰¤ r â‰¤ n. In fact

= n(n-1) (n-2) .... (r +1) - Product of r (â‰¥ 1) consecutive natural numbers can be written as quotient of two factorials.

Then m (m +1 ) (m + 2) .... (m + r -1 ) =

=.