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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" (book).
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 $B$.
Independence of a set
So lets look at some ways when is it necessary to incorporate the rule of product.