Multiplication Counting Principle

The multiplication counting principle states that the number of ways a sequence of events can occur is the product of the number of ways each individual event can occur.

Example 1:

Jack, Mary, and Tom each roll two dice. What is the number of ways in which Jack can get a total of $6,$ Mary can get a total of $4,$ and Tom can get a total of $3?$

First we must look at the individual events — let's start with Jack. Jack must get a total of $6$ from two dice. Let the possible combinations be in the form $(a, b)$ where $a$ is the number the first dice lands on and $b$ is the number the second dice lands on. The possible combinations are $(5, 1)$, $(2, 4)$, $(3, 3)$, $(4, 2)$ and $(1, 5)$ so there are $5$ ways Jack can get a total of $6.$

Similarly, the possible combinations for Mary are $(1, 3)$, $(2, 2)$, $(3, 1)$ $\rightarrow$ $3$ ways. The possible combinations for Tom are $(1, 2)$ and $(2, 1)$ $\rightarrow$ $2$ ways.

The number of ways in which Jack can get a total of $6$, Mary can get a total $4$, and Tom can get a total of $3$ is $$5\times 3\times 2 = 30.$$

Example 2:

You are planning your schedule for next year; there are $3$ Math courses, $2$ English courses and $4$ Science courses that you are eligible to enroll in. You want to take $1$ course in each subject, how many combinations of $3$ courses ($1$ from each subject) do you have to choose from?

Number the courses in each subject from $1$ to the number courses you are eligible to enroll in for that subject.

$$3 \times 2\times 4 = 24$$

Test Yourself!

What is the multiplication counting principle?

Imagine that you're making cupcakes. You've got four types of icing and three types of sprinkles. How many possible (delicious) cupcake combinations are there?

Obviously, since cupcakes are involved, this is a critical question. How can we figure this out?

The multiplication counting principle allows us to calculate the number of possible outcomes of multiple events happening at the same time. In our case, the first "event" is the icing and the second "event" is the sprinkles.

Let's come back to our cupcake problem.

(Video) The Counting Principle, Permutations, and Combinations

by Jack Walker

Jack Walker has a great video on the multiplication counting principle, permutations, and combinations.

Counting Sizes of Cartesian Products

One particular instance of the Multiplication Counting Principle is in finding the sizes of Cartesian products of sets. If $A$ and $B$ are sets, the Cartesian product of sets $A$ and $B,$ denoted $A \times B,$ is the set of all ordered pairs where the first coordinate is an element of $A$ and the second coordinate is an element of $B.$

Example: If $A = \{1, 2, 3\}$ and $B = \{4, 5\},$ then

$$A \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}.$$

At the same time, we can flip the positions of $A$ and $B$ and see that

$$B \times A = \{(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)\}.$$

In general, $A \times B$ and $B \times A$ have the same number of elements (where each element is an ordered pair), but the elements are not the same. Like in the example above, $(4, 1)$ is NOT the same as $(1, 4).$

For the next series of questions we use $X = \{0, 1, 2, 3, 4\}$ and $Y = \{-3, 5\}.$

To find the number of elements in $X \times Y,$ we have to list them out.

Finally, we can count a total of $10$ elements in $X \times Y.$ In general, the product of the sizes of $X$ and $Y$ equals the size of the Cartesian product $X \times Y.$