# Rule of Product

How to use the rule of product to find the number of ways: formula, 3 examples, and their solutions.

## Formula

If the number of ways to do A is n_{1},

the number of ways to do B is n_{2},

...

then the number of ways

to do [A and B and ...] is

N = n_{1}⋅n_{2}⋅ ... .

This is the rule of product.

(one of the counting principle)

## Example

A die shows

1, 2, 3, 4, 5, or 6.

So the number of ways

a die shows is

6.

A coin shows

head or tail.

So the number of ways

a coin shows is

2.

So the number of ways

a die and a coin shows is

6⋅2.

6⋅2 = 12

So 12 is the answer.

## Example

There are 3 spoons.

So the number of ways

to choose a spoon is

3.

There are 5 cups.

So the number of ways

to choose a cup is

5.

There are 2 dishes.

So the number of ways

to choose a dish is

2.

So the number of ways

to choose a spoon, a cup, and a dish is

3⋅5⋅2.

5⋅2 = 10

3⋅10 = 30

So 30 is the answer.

## Example

There are three cases

to get at least one multiple of 3:

at the first trial,

at the second trial,

at both trials.

It's complicated.

Then, find the number of ways of

[total] - [not getting the multiple of 3].

A die shows 1 ~ 6.

Then, from 1 ~ 6,

the multiple of 3 are

3 and 6.

The numbers that are not the multiple of 3 are

1, 2, 4, and 5.

So the number of ways

to not get the multiple of 3 is

4.

Find the total number of ways.

For the first trial,

the number of ways

a die shows is

6.

For the second trial,

the number of ways

the die shows is also

6.

So the total number of the ways is

6⋅6.

Next, subtract the number of ways

to not get the multiple of 3.

For the first trial,

the number of ways

to not get the multiple of 3 is

4.

For the second trial,

the number of ways

to not get the multiple of 3 is

4.

So, for two trials,

the number of ways

to not get the multiple of 3 is

4⋅4.

So the number of ways

to get at least one multiple of 3 is

6⋅6 - 4⋅4.

6⋅6 = 36

-4⋅4 = -16

36 - 16 = 20

So 20 is the answer.