benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let k and n be positive integers and let S={(a₁,…,a_(k))∈Z^(k)∣0≤a_(k)≤⋯≤a₁≤n,a₁+⋯+a_(k)=k} Determine, with proof, the value of …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let kk and nn be positive integers and let S={(a1,…,ak)∈Zk∣0≤ak≤⋯≤a1≤n,a1+⋯+ak=k} S=\left\{\left(a_{1}, \ldots, a_{k}\right) \in \mathbb{Z}^{k} \mid 0 \leq a_{k} \leq \cdots \leq a_{1} \leq n, a_{1}+\cdots+a_{k}=k\right\} Determine, with proof, the value of $$ \sum_{\left(a_{1}, \ldots, a_{k}\right) \in S}\binom{n}{a_{1}}\binom{a_{1}}{a_{2}} \cdots\binom{a_{k-1}}{a_{k}} $$ in terms of kk and nn, where the sum is over all kk-tuples (a1,…,ak)\left(a_{1}, \ldots, a_{k}\right) in SS.
Plain-text mathematical notation (without MathML)
Let k and n be positive integers and let S={(a₁,…,a_(k))∈Z^(k)∣0≤a_(k)≤⋯≤a₁≤n,a₁+⋯+a_(k)=k} Determine, with proof, the value of $$ \sum_{\left(a_{1}, \ldots, a_{k}\right) \in S}\binom{n}{a_{1}}\binom{a_{1}}{a_{2}} \cdots\binom{a_{k-1}}{a_{k}} $$ in terms of k and n, where the sum is over all k-tuples (a₁,…,a_(k)) in S.
Original LaTeX notation
Let $k$ and $n$ be positive integers and let $$ S=\left\{\left(a_{1}, \ldots, a_{k}\right) \in \mathbb{Z}^{k} \mid 0 \leq a_{k} \leq \cdots \leq a_{1} \leq n, a_{1}+\cdots+a_{k}=k\right\} $$ Determine, with proof, the value of $$ \sum_{\left(a_{1}, \ldots, a_{k}\right) \in S}\binom{n}{a_{1}}\binom{a_{1}}{a_{2}} \cdots\binom{a_{k-1}}{a_{k}} $$ in terms of $k$ and $n$, where the sum is over all $k$-tuples $\left(a_{1}, \ldots, a_{k}\right)$ in $S$.

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
fa3afeea-039b-5d7c-ac5a-80a6b17e5beb
task_key
test--fa3afeea-039b-5d7c-ac5a-80a6b17e5beb
task_revision_id
1
config / split
/ test

Official source

initial import