Calculate 5C2 Step by Step

Step by step

The question

Question: work out 5 C 2.

₅C₂ =

1 Write the rule

The rule for combinations: n factorial, over r factorial times n minus r, factorial.

_nC_r = n!r! × (n − r)!

Here n is 5 and r is 2.

n = 5

r = 2

2 Put the numbers in

We put 5 for n and 2 for r.

₅C₂ = 5!2! × (5 − 2)!

5 minus 2 is 3.

₅C₂ = 5!2! × 3!

3 Write out and cancel

We write every factorial out.

₅C₂ = 5 × 4 ~~× 3 × 2 × 1~~(2 × 1) × (~~3 × 2 × 1~~)

3 times 2 times 1 is on top and at the bottom.

They cancel.

What is left is 5 × 4, over 2 × 1.

₅C₂ = 5 × 4~~2 × 1~~

Why divide by 2 factorial?

The top, 5 times 4, counts in order.

One group of 2 can be put in order 2 ways.

So each group is counted 2 times.

one group of 2: 2 × 1 = 2 orders

AB, BA

Careful!

5 times 4 counts each group 2 times.

We must divide by 2.

Careful: not 5 × 4 = 20 ✗

4 Work it out

2 times 1 is 2.

4 on top and 2 at the bottom both divide by 2.

What is left is 5 × 2.

₅C₂ = 5 × 2

5 times 2 is 10.

5 × 2 = 10

So 5 C 2 is 10.

₅C₂ = 10

5 Check

Let's check: 10 groups, each in 2 orders, make 20.

10 groups × 2 orders = 20

And 5 times 4, in order, is 20.

The same.

So the answer is right!

5 × 4 = 20 ✓

Let's look back.

First: write the rule.

Then: put the numbers in.

Then: write out and cancel.

Then: work it out.

Last: check.

Answer