Factorial Examples: 41 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factorial.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The factorial of a non-negative integer $n$ , written $n!$ , is the product of all positive integers from 1 to $n$ : $n! = n \cdot (n-1) \cdots 2 \cdot 1$ .

Factorial counts the number of ways to arrange $n$ distinct objects in a row — for 3 items, there are $3! = 6$ possible orderings.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $n!$ is the product of every whole number from $n$ down to $1$ , and it counts how many ways to arrange $n$ things in a row.

Common stuck point: The procedure for factorial is the easy part; the trap is setting $0!=0$ . Asking "Am I counting the ways to arrange all $n$ distinct items, multiplying $n$ down to $1$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I counting the ways to arrange all $n$ distinct items, multiplying $n$ down to $1$ ?

Worked Examples

Example 1

easy

Compute

7!

Answer

7! = 5040

First step

Recall the factorial definition:

n! = n \times (n-1) \times \cdots \times 2 \times 1

. Write out

7!

7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1

Full solution

2
Multiply step by step: $7 \times 6 = 42$ , then $42 \times 5 = 210$ , then $210 \times 4 = 840$ , then $840 \times 3 = 2520$ .
3
Complete the product: $2520 \times 2 = 5040$ , so $7! = 5040$ .

The factorial

n!

is the product of all positive integers from

1

n

. By convention,

0! = 1

. Factorials grow extremely fast.

Example 2

medium

Simplify

\frac{10!}{8!}

Example 3

medium

Simplify

\dfrac{(n+1)!}{(n-1)!}

Example 4

medium

Compute

\binom{8}{3}

using factorials.

Example 5

medium

Simplify

\dfrac{n!\cdot (n+1)}{(n+1)!}

Example 6

hard

Compute the number of trailing zeros in

25!

Example 7

hard

Compute the largest power of

2

that divides

10!

Example 8

hard

How many distinct arrangements of the letters in 'MATHEMATICS' (11 letters) exist?

Example 9

challenge

Find the number of trailing zeros in

100!

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Evaluate

\frac{6!}{4! \cdot 2!}

Example 2

medium

Solve for

n

\frac{n!}{(n-2)!} = 30

Example 3

easy

Compute

4!

Example 4

easy

Compute

0!

Example 5

easy

Compute

\frac{6!}{5!}

Example 6

easy

Compute

1!

Example 7

easy

Compute

3!+2!

Example 8

easy

Compute

5!

Example 9

easy

How many ways can 4 distinct people line up?

Example 10

easy

Compute

\frac{7!}{7!}

Example 11

medium

Compute

\frac{8!}{6!\,2!}

Example 12

medium

Solve for

n

\frac{n!}{(n-2)!}=20

Example 13

medium

Compute

\frac{10!}{8!}

Example 14

medium

How many trailing zeros does

10!

have?

Example 15

medium

Compute

\frac{9!}{3!\,6!}

Example 16

medium

Express

7\times 6\times 5

as a ratio of factorials.

Example 17

medium

Simplify

\frac{(n+1)!}{n!}

Example 18

medium

Compute

\frac{6!}{2!\,2!\,2!}

Example 19

medium

Compute

\frac{9!}{7!}

Example 20

challenge

Show that

n!=n\times(n-1)!

and use it to compute

5!

from

4!=24

Example 21

challenge

For how many positive integers

n

n!

divisible by

100

? Find the smallest such

n

Example 22

challenge

Simplify

\frac{(n+2)!}{n!}

to a polynomial in

n

Example 23

easy

Compute

6!

Example 24

easy

How many distinct ways can

5

different books be arranged on a shelf?

Example 25

easy

Compute

\dfrac{5!}{3!2!}

Example 26

easy

In how many ways can

3

distinct friends sit in

3

chairs in a row?

Example 27

medium

Solve for

n

n!=120

Example 28

medium

How many distinct arrangements of the letters in the word 'BOOK' are there?

Example 29

medium

How many distinct

7

-letter arrangements exist for the letters in 'MISSING'?

Example 30

hard

Solve for

n

\binom{n}{2}=45

Example 31

hard

In a class of

30

, how many ways can a committee of

4

be chosen?

Example 32

hard

Solve for

n

(n+1)!=42\cdot(n-1)!

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Related Concepts

Multiplication Permutation Combination

Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplication