# Combinations (_{n}C_{r})

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

## Example 1

_{7}C_{3} means

multiply from 7 to 3 numbers (7⋅6⋅5)

and write 3! on the denominator (3⋅2⋅1)._{7}C_{3} = (7⋅6⋅5) / (3⋅2⋅1)_{n}C_{r} = _{n}P_{r} / *r*!

Permutation

## Example 2

_{7}C_{4} means

multiply from 7 to 4 numbers (7⋅6⋅5⋅4)

and write 4! on the denominator (4⋅3⋅2⋅1)._{7}C_{4} = (7⋅6⋅5⋅4) / (4⋅3⋅2⋅1)

As you can see,_{7}C_{4} = _{7}C_{3} = _{7}C_{7 - 4}.

Here's a formula that will be useful:_{n}C_{r} = _{n}C_{n - r}

## Example 3

_{9}C_{1} = 9/1 = 9._{n}C_{1} = *n*

## Example 4

_{6}C_{0} = 1._{n}C_{0} = 1

## Example 5

From 9 students,

you select 4 students.

There's **no order**.

So this is a combination problem.

So the answer is _{9}C_{4}.

## Example 6

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 _{5}C_{2}⋅_{6}C_{3}.

## Example 7

From 5 boys, you select 2 of them.

From 6 girls, you select 3 of them.

So write _{5}C_{2}⋅_{6}C_{3}.

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

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

So the answer is _{5}C_{2}⋅_{6}C_{3}⋅5!.