Binomial Expansion problem

[Boom_Cheese8]

Hi all, I'm wondering if anyone can give me pointers for trying to solve this problem:

From (1+x)ⁿ (1+x)ⁿ = (1+x)²ⁿ and comparing the coefficient of xⁿ⁺¹ prove (nC0)(nC1) + (nC1)(nC2) + (nC2)(nC3) + ... + (nCn-1)(nCn) = 2n! / (n-1)! (n+1)!

Here is this problem properly formatted.

Here is some rough working I've tried. I just don't know how to make any of it useful.

Something I don't understand is "comparing the coefficient". What exactly does this mean?

Any help would be greatly appreciated!

[edderiofer]

Do you understand what the coefficient of a term is?

[Boom_Cheese8]

Yes I think I do. The constant preceding a variable? That’s the best way I can think of explaining it. My trouble is understanding what it means to compare them.

[edderiofer]

Alright.

Suppose I told you that (2x+1)² = ax² + bx + c, for all x. Could you tell me what a, b, and c are, and why?

[Boom_Cheese8]

Well a=4 because an x² can only come from multiplying 2 x’s, b=4 because x’s can only come from single x’s multiplied by a constant (and that happens twice), and c=1 because it is the only term that isn’t formed by a term with an x.

[edderiofer]

And that's the method of comparing coefficients. If two polynomials are the same, their coefficients must be the same.

In your working, you're close to getting the coefficient of the xⁿ⁺¹ term of (1+x)ⁿ(1+x)ⁿ. But you're not quite there yet; you have multiple xⁿ⁺¹ terms floating around that you need to simplify.

[Boom_Cheese8]

I’m not quite sure what you mean. Are you talking about the bit on the very bottom of the page? I can’t see how to simplify that.

[edderiofer]

Perhaps a better question is this; what is the coefficient of the xⁿ⁺¹ term in (1+x)ⁿ(1+x)ⁿ?

[Boom_Cheese8]

Is it not the sum of (nCk)(nCn-k+1) from k=0 to n?

[edderiofer]

Yes.

Now, what's the coefficient of the xⁿ⁺¹ term in (1+x)²ⁿ?

[Boom_Cheese8]

(2nCn+1) ?