# Factorial Questions with Solutions

Tutorial on evaluating and simplifying expressions with factorial notation are presented.

 

### Definition of Factorial

Let $$n$$ be a positive integer. $$n$$ factorial, written $$n!$$, is defined by
$n! = 1 \times 2 \times 3 \times ... (n - 1) \times n$
The special case when $$n = 0$$, $0! = 1$

### Question 1

Evaluate the following expressions:
1. $$4!$$
2. $$5! \times 5!$$
3. $$3! \times 0!$$
4. $$\dfrac{ 4!}{0!}$$
5. $$\dfrac{6!}{2! \times 4!}$$

### Solution to Question 1

1. $$4! = 1 \times 2 \times 3 \times 4 = 24$$
2. $$5! \times 5! = (5!)^2 = (1 \times 2 \times 3 \times 4 \times 5)^2 = 120^2 = 14400$$
3. $$3! \times 0! = (1 \times 2 \times 3) \times 1 = 6$$
4. $$\dfrac{ 4!}{0!} = \dfrac{1 \times 2 \times 3 \times 4}{1} = 24$$
5. $$\dfrac{6!}{2! \times 4!}$$
$$= \dfrac{1 \times 2 \times 3 \times 4 \times 5 \times 6}{ (1 \times 2 ) \times ( 1 \times 2 \times 3 \times 4) }$$
$$= 15$$

### Question 2

Simplify the following expressions:
1. $$\dfrac{(n + 2)!}{n!}$$
2. $$\dfrac{(2n + 2)!}{(2n)!}$$
3. $$\dfrac{(n - 1)!}{(n + 1)!}$$

### Solution to Question 2

1. Expand the factorials
$$\dfrac{(n + 2)!}{n!}$$
$$= \dfrac{ 1 \times 2 \times ... \times n \times (n + 1) \times (n + 2) } {1 \times 2 \times ...\times n }$$
Cancel common factors, in the numerator and denominator, and simplify to obtain
$$= (n + 1)(n + 2)$$
2. Expand the factorials
$$\dfrac{(2n + 2)!}{(2n)!}$$
$$= \dfrac{ 1 \times 2 \times 3...(2n) \times (2n + 1) \times (2n + 2) } {1 \times 2 \times 3...2n}$$
Cancel common factors, in the numerator and denominator, and simplify to obtain
$$= (2n + 1) \times (2n + 2)$$
3. Expand the factorials
$$\dfrac{(n - 1)!}{(n + 1)!}$$
$$= \dfrac {1 \times 2 \times 3...(n - 1) }{1 \times 2 \times 3...(n - 1) \times n \times (n + 1)}$$
Cancel common factors in the numerator and denominator and simplify to obtain
$$= \dfrac{1}{n(n+1)}$$

### Exercises

a) Evaluate $$\dfrac{(10! / 5!)}{10}$$
b) Simplify $$\dfrac{(n + 1)!}{n!}$$

### Answers to above exercises

a) $$\dfrac{(10! / 5!)}{10} = 3024$$
b) $$\dfrac{(n + 1)!}{n!} = n +1$$

### More References and links

elementary statistics and probabilities.
Factorial Calculator to calculate the factorial of a positive integer.