# Product Rule of Counting

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?

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.

• The question said different offices. If there were the possibility of sharing, it would be $(12)(12)$. – André Nicolas May 2 '15 at 15:23
• Thanks a lot ! very foolish of me to miss that word. Thanks again :) – Teja713 May 2 '15 at 15:25
• 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 $(12)(11)$. – André Nicolas May 2 '15 at 15:38