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.

#NumberTheory

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

A Former Brilliant Member - 10 months ago

I see

Odin Wang - 10 months ago

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

James Watson - 10 months, 1 week ago

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

Odin Wang - 10 months, 1 week ago

MathHistory

Odin Wang - 10 months ago

In India it is known as Meru Prastara

A Former Brilliant Member - 10 months ago

Everywhere it is different unless it isn't

Odin Wang - 9 months, 4 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$