Combinatorics

[hollow topaz]

What do you understand? What have you tried?

[upper portal]

I think it can be solved by binomial expansion
I've tried to reach to the left side of the equations by expanding (x+y)^n.

I've also tried to prove it for a determined value for n (for example n = 4 or n = 5). it seems right but don't know where to start the proof

[hollow topaz]

The title says combinatorial proof. Do you just mean that the topic is combinatorics, or do you actually need a combinatorial proof?

Expanding it would work, but that would give you an algebraic proof. (It's also much harder.)

The combinatorial method is to pose a new question that is answered by both sides of the equation, and therefore they must be the same.

[upper portal]

It's my mistake I could use a better topic
I meant the topic is combinatorics

[hollow topaz]

Oh. So you haven't learned combinatorial proofs?

[upper portal]

Combinatorics

[hollow topaz]

Heh, I just noticed the typo that the sum should be k=1.

[upper portal]

Yep, you're right let me close the post re-post with the right equations

[hollow topaz]

Wait

It's fine

Unless this is actually the wrong question.

[upper portal]

it's right equations

\sum_{k=0}^{n}\binom{n}{k}\binom{2n}{k}=\binom{3n}{n}

[surreal yarrow]

Soroosh
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[hollow topaz]

Oh, the typo doesn't matter, lol

I understood the question without even noticing

To do this algebraically, you need to write binom(n,k) using factorials

Same with the other one

[upper portal]

I've tried it.
it's get really messy by proving in form factorials and hard to follow

[hollow topaz]

Show your work

[upper portal]

I rewrite it, it seems it leads me to a really messy equations which is hard to simplify. and it if continue the factorial farther and taking common denominator and .... it really get messy (really no space will be left on the whiteboard, lol)

[hollow topaz]

One of those should be 2n-k

[upper portal]

yeah you're right i made a mistake let me retry it

[hollow topaz]

I'm not sure that expansion the sum with a ... is helpful

This algebra is really messy and I'm not sure how it's going to get better... That could just be a blind spot on my part

[upper portal]

ThankU
it really get worse after i fix the mistake:))
let me try if binomial theorem can help

[hollow topaz]

I'm really surprised that you haven't learnt combinatorial proofs. This is a classic one. The combinatorial proof is two lines.

[upper portal]

Can you introduce text book on it? I haven't seen anything about it in the book that professor uses as the reference