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Suppose that there are 9 faculty members in the mathematics department and 11 in the computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members from the mathematics department and four from the computer science department?

 

I am always very confused as to where to use Product rule and where to use Sum rule.

How to decide on Product rule or Sum rule? Please use some basic example for both the cases.

 

2 Answers

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Rule of sum-If there are n choices for one event and m choices for another event, both cannot occur at the same time then there are n+m choices for one event

Rule of product--If there are n choices for one event and m choices for another event,  then there are n*m choices for both the event to occur

in the above question

selecting three faculty members from the mathematics department = 9C3

selecting four faculty members from the computer department= 11C4

therefore from the product rule= 9C3*11C4

                                                 =84*330

                                                  =27720 ways
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By the product rule, the answer is the product of the number of 3 – combinations of set with nine elements and the number of 4- combinations of set with 11 elements. The number of ways to select the committee is:

     C(9,3)*C(11,4) = (9!/3!*6!)  *  (11!/4!*7!) = 84 * 330 = 27,720 ways

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