# Tagged Questions

Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### Complex Analysis proof of multinomial expression

I've recently come across the following identity $$\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$$ A nice complex analysis proof (by Felix Marin, here) follows as: ...
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### Closed form of $\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$

Is there a closed form for: $$\sum_{k=1}^{n}\binom{n}{k} h^{(n-k)}(0)f^{(k-1)}(0)$$ where: $$h(x)=(1-x)^{\alpha}(A-Bx)^{\frac{1}{\gamma}-\alpha}$$ and ...
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### How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
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### Verification of a Combinatorial Identity

I have a challenge for you combinatorial mathematicians. Is anyone willing to verify the following combinatorial identity? ...
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### Partial sum of ${A \choose i} {B\choose n-i}$, when $B=-1$?

It's easy to see that $$\sum_i {A\choose i} {B\choose n-i} = {A+B\choose n}$$ since when we choose $n$ things out of $A+B$, some ($i$ of them) are in the $A$ and the rest are in the $B$. Is there ...
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### Limit involving the totient function and combination

Do you think the following limits are correct? $\displaystyle\lim_{d\to\infty}\frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{\varphi(N)^d}=0$ ...
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### ${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r},$$ which can be proved combinatorically whether one particular element (among the $n$) ...