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There are $k$ varieties of objects, each variety consisting of the same number of objects. These objects are drawn one at a time and replaced before the next drawing. What is the probability that $n$ and no fewer drawings will be required to produce objects of all varieties?

The back of my book says the answer is $\dfrac{ \sum (-1)^r~~~ ^{k-1}C_r(k-r-1)^{n-1}} { k^{n-1} }$

Attempt: It is evident that $n > k.$ The problem wants that $n$ and no fewer drawings will be required to produce objects of all varieties.

Thus, there exists at least one variety $V_i, ~1 \le i \le k$ such that all objects from $V_i$ are drawn together in the end only.

I am not sure how to proceed ahead with the inclusion-exclusion principle. Could someone please give a direction. Thanks a lot for your help!

MathMan
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@eyeballfrog said $n$'th draw must be the last varieties. Now it like in $n-1$ draw we have $k-1$ varieties without restriction of "And No Fewer Drawing"!

Now drawing $k-1$ varieties is equivalent to skip drawing $1$ variety. If $A_i$ is to skip $i$'th variety, we should compute $|\cup_i A_i|$. By Inclusion-Exclusion Principle we get the nominator of that formula.

Notice $^{k-1}C_r$ is the number of choices of $r$ varieties (to be skipped) from $k-1$ ones.

And $(k-r-1)^{n-1}$ is the number of drawings don't draw those $r$ skipped varieties.

Ali Ashja'
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  • Could you be a little more detailed , please – MathMan Oct 28 '19 at 14:35
  • @MathMan I Explained it. – Ali Ashja' Oct 29 '19 at 16:34
  • Thanks. Could you also tell me How to explain the $(-1)^r$ and the summation sign? – MathMan Oct 29 '19 at 16:39
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    They came from Inc-Exc Principle formula. If you aren't familiar with it, see https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle – Ali Ashja' Oct 29 '19 at 16:44
  • Shouldn't $^{k-1}C_r$ be the number of choices of $r$ varieties from $k-1$ ones which were not skipped ( Since, $A_i$ is that event in which only the $i th$ variety was skipped).

    And $(k-r-1)^{n-1}$ ?

     – MathMan Oct 30 '19 at 08:23
  • If you could explain the nominator in detail, it would be really helpful – MathMan Oct 30 '19 at 09:09
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    In fact, $k-1$ ones were not skipped, now we choose $r$ ones from them to be skipped. Read Inc-Exc Principle exactly. There is no more detail. Just first try to understand Inc-Exc Principle concept. Then apply it to problem. – Ali Ashja' Oct 30 '19 at 12:33
  • got it. Thank you ! – MathMan Oct 30 '19 at 18:00