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1 year ago in Combinatorics , Number Theory By Debashis Mohapatra

Is the product of k consecutive positive integers divisible by k!?

This came up in a number theory seminar, and while the result seems intuitively plausible, I'm seeking a clear, authoritative explanation. Is this a standalone lemma or a consequence of a deeper combinatorial principle? Understanding this will help in a proof I'm developing related to binomial coefficients.

FactorialNotation Consecutiveintegers

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By Joshna Answered 1 year ago

This isn't just a handy trick; it's a fundamental combinatorial identity in disguise. I've used this property countless times. The cleanest explanation is to recognize that the binomial coefficient C(n, k) = n!/(k!(n-k)!) is an integer. Your product of k consecutive integers starting at m is exactly k!  C(m+k-1, k). Since the binomial coefficient is an integer, the product must be divisible by k!. This frames it not as a curious pattern, but as a direct consequence of the definition of combinations.

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