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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 S1, S2, S3, S4 and S5 and the 3 pairs of pants by P1, P2 and P3. The different ways of dressing up is given as follows:

S1P1          S2P1              S3P1                    S4P1             S5P1  
S1P2          S2P2              S3P2                    S4P2             S5P2
S1P3          S2P3              S3P3                    S4P3             S5P3

How many ways are there? There are 5 × 3 = 15 possible ways. That is, we multiply the number of ways in which you can wear a shirt and the number of ways in which you can wear a pair of pants. Figure below gives a tree diagram on the number of ways you can dress up. The tree diagram of figure above shows we have 5 distinct groups with each group containing 3 objects. Therefore, by the sum rule, the number of ways of dressing up is 5 + 5 + 5 = 3 × 5 = 15.

 

 

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 × 3 × 4 = 60. This is because, you can wear a pair of shoes with S1 P1 in 4 ways, with S1P2 in 4 ways, with S1P3 in 4 ways and so on. That is, you have 4 branches of the tree in figure above, coming out from each of S1P1. If we denote the pairs of shoes by B1, B2, B3 and B4, then figure given below shows that with shirt S1 we have 3 × 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 × 3 × 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 × 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 × 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.




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