# 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.

1answer
7 views

### Clarifying an old question on Combinations.

So there is already a question here and I just want to clarify something in this. Link:Meaning of the question Now the accepted answer says that the answer is a power of 2.And there is an ...
1answer
1k views

### How many different chess-board situations can occur?

If you play a standard chess game on a normal $8 \cdot 8$ chess board with the usual rules: How many different "board representations" can exist? Upper bound: Well, you have 16+16 = 32 chess pieces ...
4answers
229 views

### Prove $\sum_{k=0}^{n}\frac{n!}{k!}(n-k)n^k=n^{n+1}$ for any $n\in\mathbb N$.

I want to prove the following: $$\sum_{k=0}^{n}\frac{n!}{k!}(n-k)n^k=n^{n+1}\quad\text{for any n\in\mathbb N.}$$ I tried induction and invoking the binomial theorem, to little avail. I’m looking ...
1answer
15 views

### Find the number of different magmas that have $A$ as its underlying set

I have a problem involving algebraic structures. Any help I can get here would be amazing. Problem: We have a set $A$, $\text{card} A = n$, $n \in \Bbb N$. Find the number of different magmas that ...
2answers
23 views

### What is the name of this (possibly classical) combinatorial optimization problem?

I have a finite number of sets $S_i$, each of the sets costing $p_i$ and containing some elements. Given the budget $b$ I want to select number of those sets to maximize $|S_{k_1} \cup S_{k_2} \dots|$ ...
0answers
24 views

### Proof using the product lemma

Let $S$ be the set of all finite subsets of $\mathbb N = \{1,2,3,...\}.$ We define a weight function $w$ where for a subset $X$ of $\mathbb N, w(X)$ is the sum of all the elements in $X$, with ...
0answers
25 views

### How can I know this problem talk about conditional probability , permutation, or combination ,Do you have any techniques for this . [on hold]

How can I know this problem talk about conditional probability , permutation, or combination ,Do you have any techniques for this . always my teacher gives us problem , and it does not have any ...
2answers
81 views

### $(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
1answer
174 views
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1answer
20 views

### Logic of getting a full-house of cards.

Although I understand the correct solution of finding the total number of full houses in a 52-deck of cards (finding the number of ways of selecting the first value and then finding the amount of ways ...