# 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 $\frac{n!}{r!(n-r)!}$, and they just put it in to use in the triangle. like, seriously. I need help.

Note by Odin Wang
10 months, 2 weeks ago

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Because in the nth line the kth element is equal to $\dbinom{n}{k}$. And: \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.

- 10 months, 2 weeks ago

I see

- 10 months, 2 weeks ago

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

- 10 months, 2 weeks ago

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

- 10 months, 2 weeks ago

# MathHistory

- 10 months, 2 weeks ago

In India it is known as Meru Prastara

- 10 months, 1 week ago

Everywhere it is different unless it isn't

- 10 months, 1 week ago