Rule of Product
The Rule of Product goes by other names such as the Multiplication Principle or Sequential Rule. It is a multiplication based rule one follows in certain counting procedures.
“The rule of product states that if an action can be performed by making \(A\) choices followed by \(B\) choices, then it can be performed \(AB\) ways” (Benjamin, 2009).
In other words suppose there exists a set of \(A\) choices and a set \(B\) choices, multiplying \(AB\) will be the number of ways to do \(A\) and \(B\). As a side note, notice the word “and”, this can often be a queue word for situations were one might find need for The Rule of Product.
Independence of a sets is also required when The Rule of Product is in play. That simply means that a set \(A\) and set \(B\) from earlier are completely separate.
Moreover, this rule extends to any number of sets. Meaning if we have a set \(A\), \(B\), \(C\), and \(D\) they can still be multiplied \(ABCD\).
So lets look at a simple example.
Say there is a 3 dial combination lock where each dial is completely separate and each contain the numbers 0 through 9. How many possible combinations can be made between the first, second, and third dials?
Before moving forward we should first identify our sets. We will call the first dial’s 0 through 9 digits set \(A\). We will call the second dial’s 0 through 9 digits set \(B\). We will call the third dial’s 0 through 9 digits set \(C\). Are the three sets independent? Yes they are. Note that 0 through 9 represents 10 values or choices for each dial.
Multiplying these sets together we would have \(ABC\) which would be \(10\cdot10\cdot10\) or \(10^3\) choices. So altogether are 1000 different possible combinations
Possible Exam Question
How many license plates can be made if the first 3 entries must be letters, followed by 3 numbers?