# Lesson 9.5: Factorial

## Explanation

Factorial notation is often used with permutations and probability analysis. It is the product of all of the positive integers beginning with one and progressing to the number preceding the factorial symbol. Scientific calculators will have a key marked with an exclamation point or with a letter followed by an exclamation point for this operation. Do not use the braille exclamation point for this symbol as that is to be used only as a mark of punctuation! The braille symbol for the factorial is a one-celled symbol, dots one two three four six.

### Example 1

$3!$

⠼⠒⠯

### Example 2

$3!=1\xb72\xb73$

⠼⠒⠯⠀⠨⠅⠀⠼⠂⠡⠆⠡⠒

### Example 3

$3!=3\xb72\xb71$

⠼⠒⠯⠀⠨⠅⠀⠼⠒⠡⠆⠡⠂

### Example 4

$\frac{5!}{4!}$

⠹⠢⠯⠌⠲⠯⠼

### Example 5

$\frac{n!}{\left(n-r\right)!}$

⠹⠝⠯⠌⠷⠝⠤⠗⠾⠯⠼