# Tagged Questions

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### Sums of binomial coefficients

Does anyone know something about the following sums? $$S_m(n)=\sum\limits_{k=o}^n(-1)^k{mn\choose mk}$$ Notice that $S_m(n)=0$ for odd $n$, so we only consider $S_m(2n)$. It holds that $S_0(2n)=1$, ...
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### The sum of binomial coefficients up to $k\le n/4$ does not exceed the $k$th coefficient

How would you prove the following (for when $k\leq\frac{n}{4}$)? $$\sum_{i=0}^{k-1} \binom ni \le \binom nk$$
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### Transformation of a sum

I want to prove the following or a similar result: For $1\le k \le n$ \begin{align}&1-\sum\limits_{j=k+1}^n\binom nj(1-x)^jx^{n-j}~~~~~~(1)\\ ...
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### Show that $\sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1}$?

I can't resolve this exercise and I need a tip. $$\sum_{k=0}^{r} \binom{r-k}{m} \binom{s+k}{n} = \binom{r+s+1}{m+n+1}$$ where $n \geq s$.
### Find $a_1$ given that $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$
If $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$, then $a_1$ is .. The options are $1$, $2$, $99$ or $100$. I'm sure the problem is trivial, but I just don't understand what is meant.