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How many possible ways can we distribute 8 identical objects in 5 different boxes?

The possible answers are (A) 17820, (B) 6720, (C) 2475, (D) 1188 and (E) 495.

It seems like the correct answer should be (E) 495.

But I tried $\frac{n!}{(n-p)!} = \frac{8!}{(8-5)!}$ which equals 6720 (B).

Which one is the correct answer?

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  • $\begingroup$ Why do you think it's 495? $\endgroup$ – Carl Schildkraut Sep 26 '16 at 23:50
  • $\begingroup$ It's the test's "official answer" for this question $\endgroup$ – Hancap Sep 26 '16 at 23:53
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It should be \begin{align} \binom{8+4}{4}= 495. \end{align} Suppose you have 5 distinct boxes, then there are four walls. Hence if you line up the balls and the walls, then there are 12 choose 4 ways to put the walls between the balls.

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  • $\begingroup$ I see, so it should be 12!/(4!*8!). The concept seems anti-intuitive to me though. $\endgroup$ – Hancap Sep 27 '16 at 0:19
  • $\begingroup$ Once you have the picture then it will be all clear. $\endgroup$ – Jacky Chong Sep 27 '16 at 0:19

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