# 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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### Minimum number of stickers required for 3x3 Rubik's Cube

Lately, the stickers on my V-Cube have been peeling off, and I became curious: what is the minimum number of stickers I would actually need in order to represent each unique configuration? I have but ...
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### Probability of Permutations/Combinations

How do you set up the formula for the probability of a permutation/combination? Question: If you have a group of candy with $2$ Snickers, $4$ Kit Kats, and $2$ Butterfingers and you take two pieces ...
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### Permutations and number of permitted combinations three percentages which must add up to 100%

is there a simple way to find the number of combinations of three percentage values with discrete step sizes which add up to 100%? Example: ...
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### permutations indistinguishable objects and groups

There is a group of 10 objects, 2 red, 3 blue and 5 green. If the 5 green objects should always be placed together, in how many ways we can put them on a line. I did this: As 5 places are occupied by ...
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### What is the probability that either 1 or 49 is in the winning numbers of a Lotto game?

In a simple Lotto game you have 49 numbers (1, ..., 49). 6 of them get drawn (without multiples, order is not important). What is the probability that either the number 1 or number 49 get drawn (but ...
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### counting hands shake

Mr. and Mrs. Brown gave a party for their friends they have not seen for a long time. Three couples came. During the party, some of the people were so happy to see each other again, that they even ...
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### Choosing subset of vertices connected to whole graph

Consider a simple graph $G$ with $n$ vertices. For any two vertices, either they are connected by an edge, or there is a third vertex which is connected to both of them by an edge. (It is possible ...
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### 52-Deck 3 Cards Drawn Possible Combinations Question

I have a HW problem I'm trying to pin down and I think I'm confusing myself... Question: In a card game w/ a standard 52 card deck, a hand is a set of 3 cards. Count the # of hands that are... a) ...
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### the least steps to remove all pebbles on table

On the table there are 100 bags of pebbles that contain 1, 2, 3, 4, ...., 100 pebbles respectively. In one step you are allowed to reduce the number of pebbles of any number of the bags as long as you ...
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### What is a good combinatorics text for someone studying for the Math GRE Subject Test?

I have NEVER taken a combinatorics course, outside of what one covers in a calculus-based probability course. I would be interested in knowing what would be a suitable combinatorics text for ...
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### Meaning of “Up to isomorphism”

I saw in my graph theory notes this statement "Up to isomorphism, there is one and only one $K_4$". What does the phrase "up to isomorphism" mean?
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### unique cube arrangments

i have received this math riddle which i cannot solve, the riddle: given a set of cubes, a unique shape is any shape that was created joining cubes sides together and does not match any previously ...
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### How many ways are there to go from $A$ to $B$?

How many paths are there to go from $A$ to $B$ in the following figure: Conditions: I'm not able to touch $C$ I can go up, down, left or right I'm not able to pass a line twice I couldn't ...
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### Density of the set of the fractional part of sufficiently large irrational numbers in the unit interval $[0,1]$

Is it true that $\forall x \notin \mathbb{Q}: x>1$, the set $A=\{ \operatorname{frac}(x^n): n \in \mathbb{N} \}$ is dense in $[0,1]$?
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### How many one to one correspondences are there from $A=\{A_1,A_2,A_3,A_4,A_5\}$ to $B=\{B_1,B_2,B_3,B_4,B_5\}$ such that…

I got this problem: Let $A=\{A_1,A_2,A_3,A_4,A_5\}$ and $B=\{B_1,B_2,B_3,B_4,B_5\}$ be two sets. How many one to one correspondences (one to one and onto functions) from $A$ to $B$ are there that ...
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### Number of spanning trees for these 2 figures

The solution to the number of spanning trees of the graph below is given by $6$ and $4 \times 4 - 1$ for Graph A and B respectively. I'm not sure how to get this. Please assist. I did ask a similar ...
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### Use of model theory in flag algebras

I need to learn about Razborov's "flag algebras" (see http://bit.ly/1u1a1NB) to solve a problem about graphs. Flag algebras are a very general new algebraic tool for studying combinatorial structures. ...
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### Number of spanning trees of this graph

The solution to the number of spanning trees of the graph below is given by $3 \times 2 \times 3 = 18$. I'm not sure how to get this. Please assist. Thanks! Notes: Just in case anyone was ...
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### Permutation question on alphabets

Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is Well I do understand some ...
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### Enumeration of integers are in increasing order which have gaps

I want to solve the following: Calculate the number of ways of selecting five distinct integers $x_1,x_2,x_3,x_4,x_5$ where $0\leq x_1 \lt x_2 \lt x_3 \lt x_4 \lt x_5 \leq 20$ I think this may ...
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### Interesting combinatorics

There is $n*n$ square grid. How many ways to fill it with $1$ and $0$ do we have, in case the sum in every row and every column should be even. The problem seems to be easy, but after some time and ...
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