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I am new to combinatorics and I'm reading it from Kenneth H.Rosen book. Under the topic Product rule of counting, this problem was given :

A new company with just two employees, Sanchez and Patel, rents a floor of a building with 12 offices. How many ways are there to assign different offices to these two employees?

The answer is 12*11=132 ways

By the definition of product rule,

Suppose that a procedure can be broken down into a sequence of two tasks. If there are n 1 ways to do the first task and for each of these ways of doing the first task, there are n 2 ways to do the second task, then there are n 1. n 2 ways to do the procedure.

So I think it should be 12(for Sanchez) * 12(for Patel) =144.

Please explain what mistake I was doing in a detailed manner. Thank You.

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  • $\begingroup$ The question said different offices. If there were the possibility of sharing, it would be $(12)(12)$. $\endgroup$
    – André Nicolas
    May 2, 2015 at 15:23
  • $\begingroup$ Thanks a lot ! very foolish of me to miss that word. Thanks again :) $\endgroup$
    – cosmos713
    May 2, 2015 at 15:25
  • $\begingroup$ You are welcome. Combinatorial problems can be very sensitive to exact wording. Here we imagined S being assigned her office first. For every way of assigning an office to S, there are $11$ ways to assign an office to P. Later there will be a more symmetric way of visualizing. There are $\binom{12}{2}$ ways to choose $2$ offices from $12$, and for each way there are $2!$ ways to assign the chosen offices, for a total of $\binom{12}{2}2!$. This simplifies to $(12)(11)$. $\endgroup$
    – André Nicolas
    May 2, 2015 at 15:38

2 Answers 2

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You count the office you've given to Sanchez as a possibility for Platel, something you shouldn't do as the question states that there are different offices.

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Sanchez and Patel cannot rent the same office. So, while Sanchez has 12 choices, Patel has only 11.

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