Rule of Product

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

Formula

Formula

If the number of ways to do A is n1,
the number of ways to do B is n2,
...
then the number of ways
to do [A and B and ...] is
N = n1⋅n2⋅ ... .

This is the rule of product.
(one of the counting principle)

Example 1

Example

Solution

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 2

Example

Solution

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 3

Example

Solution

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.