The Counting Principle

By Chandrajeet K

The counting principle works for two or more events occurring in series. It is used to find the total number of possible outcomes when two or more events occur together.

The counting principle states that:

If a first event has ‘p’ possible outcomes followed by a second event that has ‘q’ possible outcomes followed by a third event that has ‘r’ possible outcomes and so on…, then there are a total of ‘p × q × r × …’ possible outcomes for the series of events.

We simply multiply the number of possible outcomes of each of the events in the series to get the total number of outcomes for the series.

Here are some examples using the counting principle:


Example 1

Helen has 4 blouses and 3 skirts. How many different outfits consisting of one blouse and one skirt are possible?

Solution

Helen has:

4 blouses – B1, B2, B3, B4
3 skirts – S1, S2, S3

The different possible outfits consisting of one blouse and one skirt are:

1. B1, S1
2. B1, S2
3. B1, S3
4. B2, S1
5. B2, S2
6. B2, S3
7. B3, S1
8. B3, S2
9. B3, S3
10. B4, S1
11. B4, S2
12. B4, S3

So:

Helen can make 12 different outfits consisting of one blouse and one skirt.
This can easily be calculated using the counting principle as 4 × 3 = 12.


Example 2

A restaurant offers a choice of 2 salads, 4 main courses, and 3 desserts. Find the number of possible choices when you choose one item from each category.

Solution

You have a choice of 2 salads – S1, S2
You have a choice of 4 main courses – M1, M2, M3, M4
You have a choice of 3 desserts – D1, D2, D3
As in the previous example, our list looks something like this…

The different possible choices consisting of a salad, a main course, and a dessert are:

1. S1, M1, D1
2. S1, M1, D2
3. S1, M1, D3
4. S1, M2, D1
5. S1, M2, D2,
6. S1, M2, D3
7. …
8. …
9. …

I'm not going to list all of them! I'll let you write them down.

By the counting principle we know there are a total of 2 × 4 × 3 = 24 possible choices of a 3-course meal.


Example 3

A state’s license plates have 3 letters followed by 3 digits.

(a) How many different license plates are possible if letters and digits can be repeated?

(b) How many different license plates are possible if letters and digits cannot be repeated?

Solution

(a) How many different license plates are possible if letters and digits can be repeated?

There are 26 choices for letters (A to Z) and 10 choices for digits (0, 1, 2, 3 …9).

The license plates have 3 letters followed by 3 digits.

LETTER LETTER LETTER DIGIT DIGIT DIGIT

Using the Counting Principle:
26 × 26 × 26 × 10 × 10 × 10

= 17,576,000

So, 17,576,000 different license plates are possible if letters and digits can be repeated.

(b) How many different license plates are possible if letters and digits cannot be repeated?

There are 26 choices for the first letter but only 25 choices for the second, and 24 for the third.

For the digits, there are 10 choices for the first, but only 9 for the second, and 8 for the third.

SO:

Using the Counting Principle:

LETTER LETTER LETTER DIGIT DIGIT DIGIT

26 × 25 × 24 × 10 × 9 × 8
= 11,232,000

So, 11,232,000 different license plates are possible if letters and digits cannot be repeated.

Here is a question for you…

How could the state increase the total number of possible license plates if letters and digits cannot be repeated?