I have seen this proof which uses induction. It seems to be a way of proving the property $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$, instead of using this in the induction.
I have tried to start with the binomial property and then use induction hypotheses there to show h(n+1) is divisible by (n+1)!, but in reality, I was just expanding the binomial theorem and reorganizing, the induction wasn't needed there. I could not find a way to combine both induction and the binomial property in the same proof, I am wondering if the linked proof is enough to solve my question, or do I have to do something to make use of the $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$ property. If this is not the right approach then how to prove the theorem making use of both induction and the binomial property?
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alu
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Will the material in this thread help you? – Jyrki Lahtonen Nov 03 '21 at 09:07
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Okay, I'll try. – alu Nov 03 '21 at 13:29
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If the thread given by Jyrki Lahtonen isn't enough, here is another you can try: https://math.stackexchange.com/questions/12065/the-product-of-n-consecutive-integers-is-divisible-by-n-factorial?rq=1 – awkward Nov 03 '21 at 15:18