Finding value of $\sum^{10}_{k=2}\binom{k}{2}\binom{12-k}{2}$

Question

Finding value of $$\sum^{10}_{k=2}\binom{k}{2}\binom{12-k}{2}$$

Solution I tried:

$$(1+x)^{10}=1+\binom{10}{1}x+\binom{10}{2}x^2+\cdots +\binom{10}{10}x^{10}$$

$$(x+1)^{10}=x^{10}+\binom{10}{1}x^9+\binom{10}{2}x^7+\cdots +\binom{10}{10}$$

I did not find how do multiply terms such that i get my result. Help me

Answer 1

To complete the method sketched in the comments.

The answer is $\binom {13}5$.

To see this, suppose we are choosing $5$ values from $\{1,\cdots, 13\}$. Order the choices as $n_1<\cdots <n_5$. Let $n_3=k+1$. Clearly $2≤k≤10$. Given a choice of such an $n_3$ we see that we are asked to choose $2$ values from $\{1,\cdots, k\}$ and $2$ from $\{k+2,\cdots, 13\}$. There are $\binom k2$ ways to do the former and $\binom {13-(k+2)+1}2=\binom {12-k}2$ ways to do the latter. Hence the number of choices with a given $n_3$ is the product $\binom {k}2\times \binom {12-k}2$. Summing over $k$ we see that $$\binom {13}5=\sum_{k=2}^{10}\binom {k}2\times \binom {12-k}2$$ as desired.

Answer 2

Hint:

$$=\sum_{r=2}^{10}\dfrac{r(r-1)(12-r)(11-r)}4$$

Use the formulas in Geometric interpretation for sum of fourth powers

Answer 3

It is Chu–Vandermonde identity

$$\sum^{10}_{k=2}\binom{k}{2}\binom{12-k}{2}=\sum^{10}_{k=2}\binom{k}{2}\binom{12-k}{4-2}={12+1\choose 4+1}=1287.$$

Answer 4

$$S_{(10,2)} = \sum^{10}_{k=2}\binom{k}{2}\binom{12-k}{2}$$ You can generalize the question as: \begin{align} S_{(n,\,r)} &= \sum^{n}_{k=r}\binom{k}{r}\binom{(n+r)-k}{r}\\ \end{align}
Set $(n+r) = m\,$ and we get \begin{align} S_{(n,\,r)} &= \sum^{n}_{k=r}\binom{k}{r}\binom{m-k}{r}= \binom{m+1}{2r+1}\\ \end{align} (See Another form of the Chu–Vandermonde identity and this proof )


So

$$S_{(n,\,r)} = \binom{n+r+1}{2r+1}$$

In your case $$S_{(10,2)} = \binom{10+2+1}{2\cdot 2+1} = \binom{13}{5} = 1287$$