2

I'm given the following problem

An elevator starts at the basement with 8 people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number 6. In how many ways could the operator have perceived the people leaving the elevator if all people look alike to him? What if the 8 people consisted of 5 men and 3 women and the operator could tell a man from a woman?

The first part I was able to get. If there are 8 people with 6 floors, there are 5 "divisions", if you draw it out. i.e. |1|2|3|4|5|6|. There are 5 "divisions," if you exclude the edges.

This means that we can treat this like a permutation problem with letters, to my understanding. If P represents person and D represents divisions, we have PPPPPPPPDDDDD. By the counting principle, we can say that there are $$\frac{13!}{8! * 5!}$$ possibilities, given that there are 8 repeated "P"s and 5 repeated "D"s, this makes sense. This is equivalent to C(13, 8). My book tells me this is the right answer.

Now, looking at the solution for the second part, I apply the same logic. There are 5 men and 3 women. Therefore we have MMMMMWWWDDDDDD. Therefore, there are $$\frac{13!}{3! * 5! * 5!}$$

However, this gives me an incorrect result. How do I solve it correctly, and why doesn't my solution work?

ollien
  • 149
  • 1
    Suppose there were two separate carloads, one just men, one just women. – Doug M Jan 11 '18 at 23:57
  • 2
    The reason why your logic doesn't work is that it would treat a man exiting followed by a woman exiting on a floor differently than a woman exiting followed by a man exiting. If your interpretation is that the operator could in fact distinguish between a man and a woman and the order of the genders of people exiting matters to him, then your answer is correct. If your interpretation is like mine and the operator only cares about the number of each gender exiting on each floor and not the order that they exit, then approach as Doug suggests above. – JMoravitz Jan 11 '18 at 23:58
  • If you post this as an answer, I'll accept it. Thanks! – ollien Jan 12 '18 at 00:02

1 Answers1

1

Your answer doesn't work because, e.g., it distinguishes between ...DMWD... and ...DWMD..., which will appear the same to the operator.

Instead, solve the problem separately for the men and the women, then multiply.

BallBoy
  • 14,472