# The Product Rule

Suppose you have 5 shirts and 3 pairs of pants. In how many possible ways can you dress up by wearing a shirt and a pair of pants?

Let us denote the 5 shirts by S

_{1}, S

_{2}, S

_{3}, S

_{4}and S

_{5}and the 3 pairs of pants by P

_{1}, P

_{2}and P

_{3}. The different ways of dressing up is given as follows:

**S _{1}P_{1} S_{2}P_{1} S_{3}P_{1} S_{4}P_{1} S_{5}P_{1} **

S_{1}P_{2} S_{2}P_{2} S_{3}P_{2} S_{4}P_{2} S_{5}P_{2 }

S_{1}P_{3} S_{2}P_{3} S_{3}P_{3} S_{4}P_{3} S_{5}P_{3}

If in addition to 5 shirts and 3 pairs of pants you have 4 pairs of shoes, then the number of ways in which you can dress up by wearing a shirt, a pair of pants and a pair of shoes is 5 Ã— 3 Ã— 4 = 60. This is because, you can wear a pair of shoes with S

_{1}P

_{1}in 4 ways, with S

_{1}P

_{2}in 4 ways, with S

_{1}P

_{3}in 4 ways and so on. That is, you have 4 branches of the tree in figure above, coming out from each of S

_{1}P

_{1}. If we denote the pairs of shoes by B

_{1}, B

_{2}, B

_{3}and B

_{4}, then figure given below shows that with shirt S

_{1}we have 3 Ã— 4 = 12 choices of a pair of pants and a pair of shoes.

This is true for each of the 5 shirts. Thus, the number of ways in which you can dress up by wearing a shirt, a pair of pants and a pair of shoes is 5 Ã— 3 Ã— 4 = 60.The above illustration explains a general principle, called the product rule.

# The Product Rule

If an event can occur in m different ways, following which, another event can occur in n different ways, then the total number of different ways of occurrence of both the events in the given order is m Ã— n. This rule can be extended to more than two events.

The product rule is also called as the multiplication principle or the fundamental principle of counting.

**Illustration**

Suriti is taken to a toy shop which has 5 different kinds of Barbies and 4 different kinds of Teddy Bears. If Suriti is allowed to choose one of the Barbies and one of the Teddy Bears, she has 5 Ã— 4 = 20 ways to choose a Barbie and a Teddy bear.Note that in each of the above two illustrations, the first event occurs following which the second event occurs, following which the third event occurs and so on. That is, each of the given event occurs in the specified order.