14k views

### Sum of square binomial coefficients [duplicate]

Please feel free to close this is necessary as I didn't see exactly this question (some variations that I tried but didn't seem to apply. Prove: $$\sum_{k=0}^{n}{\binom{n}{k}^2}=\binom{2n}{n}$$ I ...
554 views

### Prove that:$\binom {2n}{n}$=$\sum_{r=0}^n [\binom nr ]^2$ [duplicate]

Prove that : $$\binom {2n}{n}=\sum_{r=0}^n \left[\binom nr\right]^2.$$ First of all, I tried to do in the principle of mathematical induction but I failed. Next, I expressed the binomial in algebraic ...
105 views

### Is there an expression for the sum of $\binom nr^2$ for each $n$? [duplicate]

Is there a standard expression for $$\sum_{r=0}^{n}\binom nr^2$$
194 views

### Simplify the expression - sum of squares of binomial coefficients [duplicate]

I need to simplify this thing: $$\sum_{k=0}^n{n\choose k}^2$$ I spent more than an hour thinking about it and the set of working solutions that sprang to my mind was unfortunately still empty. Then, ...
330 views

### How to prove that the sum of squared binomials equals $\binom{2n}{n}$ [duplicate]

I've stumbled upon this lemma a few times in my textbook: $$\sum_{k=0}^{n}\begin{pmatrix}n\\k\end{pmatrix}^2=\begin{pmatrix}2n\\n\end{pmatrix}$$ I've been trying to prove it, but I simply can't seem ...
153 views

### How to prove that $\sum_{i=0}^n\binom{n}{i}^2 =\binom{2n}{n}$ [duplicate]

How can I prove that $\sum_{i=0}^n\binom{n}{i}^2 =\binom{2n}{n}$ Thanks in advance!