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Number of solutions to linear equation $x_1+x_2+\dots+x_n=m$ when the domain of $x_i\ne$ domain of $x_j$
In the lecture notes of one of my previous classes, it was used that if we have an equation of the form
$$\tag{1}
x_1+x_2+\dots+x_n=m
$$
then the total number of solutions, when each $x_i$ is a non-...
- 175
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Proving an Upper Bound on the Number of Partitions of $n$ into $m$ Parts
Question
Show that
$$
p(n,m)\le\frac{1}{m!}\binom{n+\binom{m+1}{2}-1}{m-1}
$$
where $p(n, m)$ denotes the number of partitions of $n$ into exactly $m$ parts.
The above question is from Comtet'...
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