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I am not sure if this would be a proper title because I am a bit confused, but I was reading about proving Pascal's Triangle, and there was a proof on here I was following everything that was happening until the poster mentioned

$k! = \prod_{j=1}^k j$

He used this to basically reduce something like (n-k-1)!(n-k) to just (n-k) and (k-1)!k to just k!.

I've never seen this law before nor the symbol used in the formula, so I am a bit lost on what is actually going on.

Post I was looking at: Prove Pascal's Rule Algebraically

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Jude
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1 Answers1

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That big $\Pi$ is just saying multiply all the numbers from $j=1$ to $k$ together, which is the definition of $k!$

As for the reduction, $(n-k-1)!$ is multiplying all the terms up to (and including) $n-k-1$, so when we multiply by $n-k$, we're now at $(n-k)!$. Similarly for the second factorial.

smingerson
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