Pascals Triangle; who made up the combination pattern?

Here is the pascals triangle. It was invented by the Chinese, but since we are technically Europeans, we know it as Pascal's Triangle because we first knew it thanks to Pascal. But one thing I really don't understand, is why they notation combinations as n!r!(nr)!\frac{n!}{r!(n-r)!}, and they just put it in to use in the triangle. like, seriously. I need help.

Note by Odin Wang
2 months, 3 weeks ago

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Because in the nth line the kth element is equal to (nk)\dbinom{n}{k}. And: (nk)=(n1k1)+(n1k)n!k!(nk)!=(n1)!(k1)!(nk)!+(n1)!k!(nk1)!×(k1)!(nk1)!(n1)!nk(nk)=1(nk)+1knk(nk)=kk(nk)+nkk(nk)nk(nk)=nk(nk)\begin{aligned} \dbinom{n}{k}&=\dbinom{n-1}{k-1}+\dbinom{n-1}{k}\\ \cfrac{n!}{k!(n-k)!}&=\cfrac{(n-1)!}{(k-1)!(n-k)!}+\cfrac{(n-1)!}{k!(n-k-1)!}&\left \| \times\cfrac{(k-1)!(n-k-1)!}{(n-1)!}\right .\\ \cfrac{n}{k(n-k)}&=\cfrac{1}{(n-k)}+\cfrac{1}{k}\\ \cfrac{n}{k(n-k)}&=\cfrac{k}{k(n-k)}+\cfrac{n-k}{k(n-k)}\\ \cfrac{n}{k(n-k)}&=\cfrac{n}{k(n-k)} \end{aligned} So if this is true for the first elements, then it will be always true.

Páll Márton (no activity) - 2 months, 2 weeks ago

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I see

Odin Wang - 2 months, 2 weeks ago

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do you want to know what the triangle can be used for?

James Watson - 2 months, 2 weeks ago

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No thanks, I know the path trick where the Pascal’s triangle is useful.

Odin Wang - 2 months, 2 weeks ago

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MathHistory

Odin Wang - 2 months, 2 weeks ago

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In India it is known as Meru Prastara

Devbrat Dandotiya - 2 months, 1 week ago

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Everywhere it is different unless it isn't

Odin Wang - 2 months, 1 week ago

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