# Number of Ways (Math)

See how to find the number of ways

(rule of sum, rule of product, complementary event).

6 examples and their solutions.

## Rule of Sum

### Formula

N = m + n + ...

To find the number of wanted ways (N),add the possible number of ways (m, n, ...).

### Example

Numbers from 1 to 10 are given.

Find the number of ways to pick an even number.

Solution Find the number of ways to pick an even number.

2, 4, 6, 8, 10 - [1]

∴ 5 - [2]

∴ 5 - [2]

[1]

Even numbers in 1 ~ 10

[2]

There are 5 ways to get an even number.

Close

### Example

A big die and a small die are tossed once.

Find the number of ways to get a sum of 9.

(The plural of die is dice.)

⚀ ⚁ ⚂ ⚃ ⚄ ⚅

Solution Find the number of ways to get a sum of 9.

(The plural of die is dice.)

⚀ ⚁ ⚂ ⚃ ⚄ ⚅

(3, 6), (4, 5), (5, 4), (6, 3) - [1]

∴ 4

∴ 4

[1]

(Big die, Small die)

Close

### Example

Find the number of shortest paths to move from A to B.

Solution To make the shortest path,

you should move either → or ↓.

Number of ways to go to this blue point: 1

you should move either → or ↓.

Number of ways to go to this blue point: 1

↓

Number of ways to go to this blue point: 1

↓

Number of ways to go to this blue point:

1 + 1 = 2

1 + 1 = 2

↓

Number of ways to go to this blue point: 1

↓

Number of ways to go to this blue point:

1 + 2 = 3

1 + 2 = 3

↓

...

↓

∴ 14

Add the number of ways like this.

→ Number of ways to go to B: 14

→ Number of ways to go to B: 14

Close

## Rule of Product

### Formula

N(A and B) = N(A)⋅N(B)⋅...

N(A and B): Number of ways of [A and B] happening N(A): Number of ways of A happening

N(B): Number of ways of B happening

A and B don't affect each other.

### Example

A fair die and a coin is tossed once.

Find the number of results.

Solution Find the number of results.

6⋅2 = 12

Die: 1, 2, 3, 4, 5, 6

→ 6 ways

Coin: head, tail

→ 2 ways

Getting a number from a die and getting a side from a coin don't affect each other.

→ Number of results: 6⋅2

→ 6 ways

Coin: head, tail

→ 2 ways

Getting a number from a die and getting a side from a coin don't affect each other.

→ Number of results: 6⋅2

Close

### Example

3 spoons, 5 cups, 2 dishes

Find the number of ways to pick a spoon, a cup, and a dish.

Solution Find the number of ways to pick a spoon, a cup, and a dish.

3⋅5⋅2 - [1]

= 3⋅10

= 30

= 3⋅10

= 30

[1]

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

Close

## Complementary Event

### Formula

N(A) = N(total) - N(not A)

N(not A): Number of ways of A not happening (= complementary event)

In most cases, if there's [at least],

it's good to use this formula.

### Example

A fair die is tossed twice.

Find the number of ways to get at least one multiple of 3.

Solution Find the number of ways to get at least one multiple of 3.

3, 6 - [1] [2]

1, 2, 4, 5

→ 4 ways - [3]

N = 6

= 36 - 16

= 20

1, 2, 4, 5

→ 4 ways - [3]

N = 6

^{2}- 4^{2}- [4]= 36 - 16

= 20

[1]

(getting at least one multiple of 3)

= (total) - (not getting a multiple of 3 twice)

→ Find the number of (not getting a multiple of 3).

→ First find the number of getting a multiple of 3.

= (total) - (not getting a multiple of 3 twice)

→ Find the number of (not getting a multiple of 3).

→ First find the number of getting a multiple of 3.

[2]

The multiples of 3 from a die: 3, 6.

[3]

The numbers that are not a multiple of 3:

1, 2, 4, 5

→ 4 ways

1, 2, 4, 5

→ 4 ways

[4]

Number of total results (1 ~ 6, twice):

6⋅6 = 6

Number of not getting a multiple of 3 twice:

4⋅4 = 4

6⋅6 = 6

^{2}Number of not getting a multiple of 3 twice:

4⋅4 = 4

^{2}Close