# Factorial Questions with Solutions

Tutorial on evaluating and simplifying expressions with factorial notation.

### Definition of Factorial

Let n be a positive integer. n factorial, written n!, is defined by
n! = 1 � 2 � 3 � ... (n - 1) � n

The special case when n = 0,
0 factorial is given by: 0! = 1

### Question 1

Evaluate the following expressions:
1. 4!
2. 5! � 5!
3. 3! � 0!
4. 4! / 0!
5. 6! / (2! � 4!)

### Solution to Question 1

1. 4! = 1 � 2 � 3 � 4 = 24
2. 5! � 5! = (1 � 2 � 3 � 4 � 5) 2 = 120 2 = 14400
3. 3! � 0! = (1 � 2 � 3) � 1 = 6
4. 4! / 0! = (1 � 2 � 3 � 4) / 1 = 24
5. 6! / (2! � 4!)
= (1 � 2 � 3 � 4 � 5 � 6) / [ (1 � 2 ) � ( 1 � 2 � 3 � 4) ]
= 15

### Question 2

Simplify the following expressions:
1. (n + 2)! / n!
2. (2n + 2)! / 2n!
3. (n - 1)! / (n + 1)!

### Solution to Question 2

1. Expand the factorials
(n + 2)! / n!
= [ 1 � 2 � ... � n � (n + 1) � (n + 2) ] / [ 1 � 2 � ...� n ]
and simplify to obtain
= (n + 1)(n + 2)
2. Expand the factorials
(2n + 2)! / 2n!
= [ 1 � 2 � 3...(2n) � (2n + 1) � (2n + 2) ] / [ 1 � 2 � 3...2n ]
Simplify
= (2n + 1) � (2n + 2)
3. Expand the factorials
(n - 1)! / (n + 1)!
= [ 1 � 2 � 3...(n - 1) ] / [ 1 � 2 � 3...(n - 1) � n � (n + 1) ]
Simplify
= 1 / [ n � (n + 1) ]

### Exercises

a) Evaluate (10! / 5!) / 10
b) Simplify (n + 1)! / n!