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Multiplication Theorem

If there are two jobs in such a way that one of them can be done in m ways and when it is completed in any of the m ways, the second job can be completed in n ways, then the whole job can be done in m × n.

Addition Theorem

If there are two jobs in such a way that one of them can be done in m ways and the second one can be done in n ways independently, then either of the jobs can be done in (m + n) ways.

Basically, there is one point where these two theorems differ—in multiplication the job does not get completed while in addition it gets completed. In a layman’s language, we multiply the number of ways when the job has not been completed and we add the number of ways when the job has been completed.
 
Example
There are 10 girls and 15 boys in a class. In how many ways can
  1. a class representative be selected?
  2. a team of two students be chosen with one girl and one boy?
Solution
  1. A class representative can be a girl or a boy. Now, one girl can be selected from 10 girls in 10 ways (any of the girls can be selected) and one boy can be selected from 15 boys in 15 ways (any of the boys can be selected). So the ways of selecting a class representative includes either selecting a boy or a girl. So, the total number of ways of selecting a class representative =10 + 15 = 25
     
    In this case, the moment a girl gets selected, the job is completed. There are some more ways of doing this by selecting a boy. So, it is a case of addition.
  2. One girl can be chosen from 10 girls in 10 ways. Now corresponding to every selected girl, any one of the 15 boys can be selected in 15 ways.
It can be seen in the following presentation:
 
Girl selected (Assume name of the girls are G1, G2, G3,…G9, G10) – G1
 
Boy selected (Assume the names of the boys are B1, B2, B3,…B14, B15 )– B1, Or B2, or B3, or B15.
 
So, corresponding to G1, the total number of selection of a boy = 15
 
Corresponding to G2, the total number of selection of a boy = 15
 
Corresponding to G3, the total number of selection of a boy = 15
 
Corresponding to G15, the total number of selection of a boy = 15
 
So, the total number of ways of selecting a team of one boy and a girl = the total number of ways of selecting a girl × the total number of ways of selecting a boy = 10 × 15 = 150
 
In this case, just by selecting a girl or a boy, work has not been completed. So, it is a case of multiplication.
 

Another Example of Multiplication Theorem

If there are three cities A, B and C located in such a way that there are 3 roads joining A and B, and 4 roads joining B and C, then the number of ways one can travel from A to C is 3 × 4, i.e., 12.





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