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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?

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If all the people look alike to the elevator guy then it would be simply distributing 8 identical balls to 6 identical boxes and this would look like $x_1+x_2+x_3+x_4+x_5+x_6 =8$ and the solution would be $\binom{6+8-1}{8}=\binom{13}{8}=1287$

If the men and women are treated separately then the problem can be viewed as a combination of two simpler problems, i.e. distributing 5 red balls into 6 identical boxes and distributing 3 blue balls into 6 identical boxes. So, this would turn out to look like: $x_1+x_2+x_3+x_4+x_5+x_6 = 5$ corresponding to men being distributed to six floors and $y_1+y_2+y_3+y_4+y_5+y_6 = 3$ for the women. And the product of the number of solutions to the above equations gives us the final answer. And this would be $\binom{6+5-1}{5}*\binom{6+3-1}{3}=\binom{10}{5}*\binom{8}{3}=14112$
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