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Factorials

Probability · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Fill the places
  2. Write it as a factorial
  3. Multiply
  4. Check

The steps for How many ways can 4 cards be put in a row. Every solution names its steps, so you always know what comes next.

Worked examples

How many ways can 4 cards be put in a row Easy

The question

Question: How many ways can 4 cards be put in a row?

1 Fill the places

2 Write it as a factorial

So the number of ways is 4 × 3 × 2 × 1.

ways = 4 × 3 × 2 × 1

4 times 3 times 2 times 1 has a short name: 4 factorial.

We write it 4!

4 × 3 × 2 × 1 = 4!
(4 factorial)

Careful!

A card in one place cannot go in another.

So each place has one fewer.

Careful: not 4 × 4 × 4 × 4 ✗

3 Multiply

4 times 3 is 12.

12 times 2 is 24.

Times 1 changes nothing.

4 × 3 = 12
12 × 2 = 24

So there are 24 ways.

ways = 24

4 Check

Let's check the other way round: 1 times 2 is 2, then 6, then 24.

The same.

So the answer is right!

1 × 2 × 3 × 4: 2, 6, 24 ✓

Let's look back.

First: fill the places.

Then: write it as a factorial.

Then: multiply.

Last: check.

Answer
ways = 24

▶ Watch it solved

Calculate 7! Medium

The question

Question: work out 7 factorial.

7! =

1 Write it out

7 factorial means 7 times every whole number below it, down to 1.

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Factorial: multiply down to 1.

n! = n × (n − 1) × … × 1

Careful!

Factorial means times, not plus.

Careful: not 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 ✗

2 Multiply

7 times 6 is 42.

42 times 5 is 210.

210 times 4 is 840.

840 times 3 is 2520.

2520 times 2 is 5040.

Times 1 changes nothing.

7 × 6 = 42
42 × 5 = 210
210 × 4 = 840
840 × 3 = 2520
2520 × 2 = 5040

So 7 factorial is 5040.

7! = 5040

3 Check

Let's check the other way round: 1 times 2 is 2, then 6, then 24, then 120, then 720, then 5040.

The same.

So the answer is right!

1 × 2 × 3 × 4 × 5 × 6 × 7: 2, 6, 24, 120, 720, 5040 ✓

Let's look back.

First: write it out.

Then: multiply.

Last: check.

Answer
7! = 5040

▶ Watch it solved

Calculate 9!/7! Harder

The question

Question: work out 9 factorial over 7 factorial.

9!7! =

1 Write out top and bottom

We write both factorials out.

9 factorial on top, 7 factorial at the bottom.

9!7! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 17 × 6 × 5 × 4 × 3 × 2 × 1

2 Cancel

7 times 6 times 5 times 4 times 3 times 2 times 1 is on top and at the bottom.

A top and a bottom divided by the same number: they cancel.

Careful!

We never divide 9 by 7.

We cancel the numbers that are the same.

Careful: cancel the same numbers

What is left is 9 × 8.

9!7! = 9 × 8

3 Multiply

9 times 8 is 72.

9 × 8 = 72

So it is 72.

9!7! = 72

4 Check

Let's check: 9 factorial is 9 times 8, times 7 factorial.

9! = 9 × 8 × 7!

So dividing by 7 factorial leaves 9 times 8: 72.

So the answer is right!

9! ÷ 7! → 9 × 8 = 72

Let's look back.

First: write out top and bottom.

Then: cancel.

Then: multiply.

Last: check.

Answer
9!7! = 72

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. how many ways can 4 cards be put in a row
    Answer
    ways = 24
  2. 4!
    Answer
    4! = 24
  3. 5!
    Answer
    5! = 120
  4. 9!/8!
    Answer
    9!8! = 9
  5. how many ways can 4 friends be lined up in a row
    Answer
    ways = 24
  6. 6!
    Answer
    6! = 720

Medium

  1. 7!
    Answer
    7! = 5040
  2. how many ways can 5 books be put in a row
    Answer
    ways = 120
  3. how many ways can 6 runners be lined up in a row
    Answer
    ways = 720
  4. 5!/3!
    Answer
    5!3! = 20
  5. 6!/4!
    Answer
    6!4! = 30
  6. 7!/5!
    Answer
    7!5! = 42
  7. 8!/6!
    Answer
    8!6! = 56

Harder

  1. 9!/7!
    Answer
    9!7! = 72
  2. 10!/8!
    Answer
    10!8! = 90
  3. 8!/5!
    Answer
    8!5! = 336
  4. 6!/3!
    Answer
    6!3! = 120
  5. 10!/7!
    Answer
    10!7! = 720
  6. how many ways can 7 photos be arranged in a row
    Answer
    ways = 5040
  7. 8!
    Answer
    8! = 40320