Combinations (nCr)

Combinations (nCr)

How to solve combination problems: formula, examples, and their solutions.

Example 1

Evaluate the given number: 7C3.

7C3 means
multiply from 7 to 3 numbers (7⋅6⋅5)
and write 3! on the denominator (3⋅2⋅1).

7C3 = (7⋅6⋅5) / (3⋅2⋅1)

nCr = nPr / r!

Permutation

Example 2

Evaluate the given number: 7C4.

7C4 means
multiply from 7 to 4 numbers (7⋅6⋅5⋅4)
and write 4! on the denominator (4⋅3⋅2⋅1).

7C4 = (7⋅6⋅5⋅4) / (4⋅3⋅2⋅1)

As you can see,
7C4 = 7C3 = 7C7 - 4.

Here's a formula that will be useful:
nCr = nCn - r

Example 3

Evaluate the given number: 9C1.

9C1 = 9/1 = 9.

nC1 = n

Example 4

Evaluate the given number: 6C0.

6C0 = 1.

nC0 = 1

Example 5

There are 9 students in a class. How many ways are there to select 4 students?

From 9 students,
you select 4 students.
There's no order.

So this is a combination problem.

So the answer is 9C4.

Example 6

There are 5 boys and 6 girls in a classroom. How many ways are there to select 2 boys and 3 girls?

From 5 boys, you select 2 of them.
From 6 girls, you select 3 of them.
There's no order.

So this is a combination problem.

So the answer is 5C26C3.

Example 7

There are 5 boys and 6 girls in a classroom. How many ways are there to select 2 boys and 3 girls and arrange them in a row?

From 5 boys, you select 2 of them.
From 6 girls, you select 3 of them.

So write 5C26C3.

Then you arrange these 2 boys and 3 girls
in order.

So multiply 5! (2 boys + 3 girls).

So the answer is 5C26C3⋅5!.