# Linked Questions

2answers
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 ...
3answers
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 ...
3answers
105 views

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

Is there a standard expression for $$\sum_{r=0}^{n}\binom nr^2$$
2answers
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, ...
4answers
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 ...
2answers
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!
2answers
68 views

### What is the proof that the two below are equal? [duplicate]

Can you help me to prove the following equality ?
6answers
50k views

I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$. I already know the logical Proof: $${n \choose k}^2 = {... 6answers 12k views ### Proof of a binomial identity \sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}. Prove that$$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$The exercise provides the following hint: \,\,\displaystyle{n \choose k}={n\choose n-k}. Any help? 3answers 2k views ### How to get {n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y} I found this in my test book, any hints? Given$${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$$Then find the value of x and y in n. According to the answer ... 4answers 4k views ### Proving  \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n  without induction [duplicate] I have to prove that:$$ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $$I don't want a complete solution, but only a hint. 3answers 3k views ### N women and N men. Groups of pairs. So we have N men and N women. We are creating groups of pairs. It is not necessary to use every man and woman. How many groups can we make ? So if we number them from 1 to N - let W_{1} be ... 6answers 112 views ### Prove that 2^n(n!)^2 \leq (2n)! Prove that 2^n(n!)^2 \leq (2n)! One can also use the following result to prove the above: 2 · 6 · 10 · 14 · · · · · (4n − 2) = \frac{(2n)!}{ n!}. The above relation gives, (2n)!=2^n n! (1.3.5..... 2answers 427 views ### Computing C_0^2+C_1^2+C_2^2+C_3^2+ \cdots +C_n^2 [duplicate] If C_k denotes binomial coefficient of choosing k objects from a set of n objects how to calculate this:$$C_0^2+C_1^2+C_2^2+C_3^2+\cdots +C_n^2$$2answers 76 views ### Calculate \sum_{i=0}^{n}i^2 \cdot \binom{n}{i}^2  Calculate$$\sum_{i=0}^{n}i^2 \cdot \binom{n}{i}^2 $$I tried to do this via combinatorics interpretation. So we want$$ \langle A,a,B,b \rangle $$such that$$ A \subset \left\{ 1,...,n \right\} \\ ...

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