# Combination (Math)

See how to solve a combination (_{n}C_{r}).

4 examples and their solutions.

## Combination

### Formula

r numbers

Meaning: From n things, _{n}C_{r}= n⋅(n - 1)⋅(n - 2)⋅ ...r!pick r things.

(no arrangement)

Permutation

Factorial

_{n}C

_{r}=

_{n}C

_{n - r}- [1]

_{n}C

_{1}= n - [2]

_{n}C

_{0}=

_{n}C

_{n}= 1 - [3]

[1]

From n things,

(number of ways to pick r things)

= (number of ways to leave (n - r) things)

(number of ways to pick r things)

= (number of ways to leave (n - r) things)

[2]

From n things, pick 1 thing

→ n ways

→ n ways

[3]

From n things,

(number of ways to pick 0 things)

= (number of ways to leave n things)

→ Don't do anything.

→ 1 way

(number of ways to pick 0 things)

= (number of ways to leave n things)

→ Don't do anything.

→ 1 way

### Example

_{7}C

_{3}

_{7}C

_{3}

= 7⋅6⋅53!

= 7⋅6⋅53⋅2⋅1

= 7⋅5

= 35

Close

### Example

_{8}C

_{6}

_{8}C

_{6}

=

_{8}C

_{2}- [1]

= 8⋅72!

= 8⋅72⋅1

= 4⋅7

= 28

[1]

_{8}C

_{6}=

_{8}C

_{8 - 6}

=

_{8}C

_{2}

Close

### Example

5 cups, 6 spoons

Find the number of ways to choose 2 cups and 3 spoons.

Solution Find the number of ways to choose 2 cups and 3 spoons.

N =

= 5⋅42!⋅6⋅5⋅43!

= 5⋅42⋅1⋅6⋅5⋅43⋅2⋅1

= 5⋅2 ⋅ 5⋅4

= 10⋅20

= 200

_{5}C_{2}⋅_{6}C_{3}- [1]= 5⋅42!⋅6⋅5⋅43!

= 5⋅42⋅1⋅6⋅5⋅43⋅2⋅1

= 5⋅2 ⋅ 5⋅4

= 10⋅20

= 200

[1]

From 5 cups,

pick 2 cups.

→

From 6 spoons,

pick 3 spoons.

→

Picking a cup and picking a spoon don't affect each other.

→ Multiply these two.

Number of Ways (Math)

pick 2 cups.

→

_{5}C_{2}From 6 spoons,

pick 3 spoons.

→

_{6}C_{3}Picking a cup and picking a spoon don't affect each other.

→ Multiply these two.

Number of Ways (Math)

Close

### Example

3 letters: a, b, c

4 numbers: 1, 2, 3, 4

Find the number of ways to choose 2 letters and 2 numbers

and arrange them in a row.

Solution 4 numbers: 1, 2, 3, 4

Find the number of ways to choose 2 letters and 2 numbers

and arrange them in a row.

N =

=

= 3⋅4⋅32!⋅4!

= 3⋅4⋅32⋅1⋅4⋅3⋅2⋅1

= 3⋅2⋅3⋅4⋅3⋅2

= 6⋅12⋅6

= 72⋅6

= 432

_{3}C_{2}⋅_{4}C_{2}⋅4! - [1] [2]=

_{3}C_{1}⋅_{4}C_{2}⋅4!= 3⋅4⋅32!⋅4!

= 3⋅4⋅32⋅1⋅4⋅3⋅2⋅1

= 3⋅2⋅3⋅4⋅3⋅2

= 6⋅12⋅6

= 72⋅6

= 432

[1]

From 3 letters,

pick 2 letters.

→

From 4 numbers,

pick 2 numbers.

→

Picking letters and picking numbers don't affect each other.

→ Multiply these two.

pick 2 letters.

→

_{3}C_{2}From 4 numbers,

pick 2 numbers.

→

_{4}C_{2}Picking letters and picking numbers don't affect each other.

→ Multiply these two.

[2]

Close