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.

  • $\begingroup$ The question said different offices. If there were the possibility of sharing, it would be $(12)(12)$. $\endgroup$ Commented May 2, 2015 at 15:23
  • $\begingroup$ Thanks a lot ! very foolish of me to miss that word. Thanks again :) $\endgroup$
    – cosmos713
    Commented 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 $(12)(11)$. $\endgroup$ Commented May 2, 2015 at 15:38

2 Answers 2


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.


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


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