<s>[INST] How to calculate $\sum_{m=1}^{N}\binom{m+k-1}{m}$. What would be a simplified formula for $\displaystyle \sum_{m=1}^{N}\binom{m+k-1}{m}$ for a given number $k$ and any number $N$?
 [/INST] Use the following formula:
$$\displaystyle\binom{m+k-1}{m}=\binom{m+k}{m}-\binom{m+k-1}{m-1}$$
Proof:
\begin{align}
\binom{m+k}{m}-\binom{m+k-1}{m-1}&=\dfrac{(m+k)!}{m!\:k!}-\dfrac{(m+k-1)!}{(m-1)!\:k!}
\\
&=\dfrac{(m+k-1)!}{(m-1)!\:k!}\left(\dfrac{m+k}{m}-1\right)
\\
&=\dfrac{(m+k-1)!}{m!\:(k-1)!}
\\
&=\binom{m+k-1}{m}
\end{align}
So
\begin{align}
\sum \limits_{m=1}^{N}\binom{m+k-1}{m}&=\sum \limits_{m=1}^{N}(\binom{m+k}{m}-\binom{m+k-1}{m-1})
\\
&=\sum \limits_{m=1}^{N}\binom{m+k}{m}-\sum \limits_{m=1}^{N}\binom{m+k-1}{m-1}
\\
&=\sum \limits_{m=1}^{N}\binom{m+k}{m}-\sum \limits_{m=0}^{N-1}\binom{m+k}{m}
\\
&=\sum \limits_{m=1}^{N}\binom{m+k}{m}-\sum \limits_{m=1}^{N-1}\binom{m+k}{m}-1
\\
&=\binom{N+k}{N}-1
\end{align}
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