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Concept

Factorial

For any integer

n

which is positive, the product of all integers from

1

to

n

is called

n

factorial, denoted as

n!

$n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot (n - 2) \cdot (n - 1) \cdot n$

The factorial of $n$ can alternatively be defined as the product of all the natural numbers less than $n .$ Therefore, it is also very common to see the factors listed in descending order.

$n! = n \cdot (n - 1) \cdot (n - 2) \cdot \dots \cdot 3 \cdot 2 \cdot 1$

The table below shows the factorial of some numbers. By rewriting the expressions, it can be seen that a special pattern emerges.

$n$	$n!$	Simplify	Rewrite
$1$	$1$	$1$	$1$
$2$	$2 \cdot 1$	$2$	$2 \cdot 1!$
$3$	$3 \cdot 2 \cdot 1$	$6$	$3 \cdot 2!$
$4$	$4 \cdot 3 \cdot 2 \cdot 1$	$24$	$4 \cdot 3!$
$5$	$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$	$120$	$5 \cdot 4!$
$⋮$	$⋮$	$⋮$	$⋮$

From the table above, it can be concluded that the factorial of a number follows a recursive rule.

$1! = 1, n! = n \cdot (n - 1)!$

The definitions discussed above apply to positive integers, but it may be helpful to know the value of $0$ factorial. By convention, $0! = 1 .$

$0! = 1$

Extra

Why is

0!

defined to be

1

?

This may seem strange since, by the definition of factorial, this means that the product of multiplying no numbers is equal to

1 .

Nevertheless, defining

0!

as

1

is necessary for the usage of the factorial of a number. For example, if the recursive property is used with

n = 1,

the following result is obtained.

n! 1! 1! 1 = n \cdot (n - 1)! = 1 \cdot (1 - 1)! = 1 \cdot 0! = 1 \cdot 0!

The last equality,

1 = 1 \cdot 0!,

only holds true if

0! = 1 .

In a similar way, there are many applications of the factorial in different areas of mathematics that only make sense if

0! = 1 .

For example, the number of ways to choose groups of

r

elements from a set of

n

elements — combinations of

n

r

— is given by the formula shown below.

_{n} C_{r} = \frac{n !}{( n - r ) ! n !}

In particular, there is only one way to make groups of

n

elements from a set having

n

elements. This is by making a group using all the elements. This is what happens when substituting

n = n

and

r = n .

_{n} C_{n} 11 = \frac{n !}{( n - n ) ! n !} = \frac{n !}{0 ! n !} = \frac{1}{0 !}

Once more, for everything to be consistent, the only value that can be assigned to

0!

is

1 .

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