# Tagged Questions

For questions about the study of finite or countable discrete structures, specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

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### The locker puzzle - predetermined strategy

The question is related to the famous locker puzzle: The director of a prison offers 100 prisoners on death row, which are numbered from 1 to 100, a last chance. In a room there is a cupboard with ...
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### $\binom{n}{k}$ modulo prime power for large $n$ and small $k$

I have to compute several value of $\binom{n}{k}$ mod $p^a$ for prime $p$ over a range of $k$, where $n$ is large and fixed, and $k$ is small and dynamic. Is there a way to speed the process up? If I ...
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### $32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity \begin{align*} ...
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### Probability of hitting a number $Ib$ (rare case)

Consider a set $S$ of $N^{3/2}$ numbers. Fix a collection $T$ of $N^{\frac{1}{2}}$ numbers. With every trial, we have the freedom to choose $N^{1-\epsilon}$ of them at a time without overlapping. My ...
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### Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$

I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone ...
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### Generalization to: n children at a round table swap places with their neighbors

$n > 3$ children occupy the places $0,..., n-1$ mod $(n)$ , so that every place is occupied by exactly 1 child. The original problem states: Now the children are allowed to swap places, ...
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### Is the limit $\lim_{n\to \infty}\left(\sum^{n}_{r=0} \binom{n}{r}\big/{n^{r}(r+3)}\right)$ rational or irrational?

How can I prove that the result of the following limit is rational/irrational?$$\lim_{n\to \infty}\left(\sum^{n}_{r=0} \frac{\binom{n}{r}}{n^{r}(r+3)}\right)$$ Would solving this limit satisfy? How ...
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### Probability of visiting $4$ cities

On her vacations Veena visits four cities $(A, B, C\ \text{and}\ D)$ in a random order. What is the probability that she visits (i) $A$ before $B$? (ii) $A$ before $B$ and $B$ ...
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### Clarification needed to understand elementary combinatorics problem

10 objects are randomly distributed among 3 boxes. What is the probability to have 6 objects in one of the boxes, 3 in another one and a single object in the remaining third box. My solution is ...