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How can we prove that the product of $n$ consecutive integers is divisible by $n$ factorial?

Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that "does not use the properties of binomial coefficients". Please post answers in said newer thread so that this incorrectly-posed question may be closed as a duplicate.

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    $$\frac1{n!}\prod_{k=0}^{n-1}(j+k)=\binom{n+j-1}{n}$$2010-11-27
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    @J.M. I didn't realize when I was writing the answer that you put this comment. I guess it would be a nice feature to have the page let you know when a new comment has been added while you're either writing a comment or an answer, just as is done with new answers.2010-11-27
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    I wish to obtain a proof which does not use the properties of binomial coefficients.2010-11-27
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    @Adrián: It's cool; you elaborated a bit more than I would've, so I've upvoted your answer already. @Paulo: you mean a combinatorial argument or something?2010-11-27
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    possible duplicate of [Proof that a Combination is an integer](http://math.stackexchange.com/questions/11601/proof-that-a-combination-is-an-integer)2010-11-27
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    @Paolo: Please don't open dupes. Please edit the original question to indicate what you desire. It is ok to edit the questions.2010-11-27
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    @Paolo: by Dupe I meant the other question you opened. Not the one Bill is referring to in his previous comment here.2010-11-27
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    @Paolo: Please don't change the question at this point since it will invalidate most of the prior answers here. Hopefully a moderator will do the right thing and merge the answers to the constrained question into one question and close the other as a duplicate.2010-11-27
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    I have flagged this for mod attention, to merge with the other one.2010-11-27
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    @Moron: The mods may well be fed up with turkeys, so be patient.2010-11-27
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    @Bill: Huh? Flagging for mod attention is the only way to ensure getting mod attention!2010-11-27
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    @Moron: It's a joke. If you're not familiar with US holidays then google "turkey day".2010-11-27

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