Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially 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.

38,462 questions
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Find invalid values of $Y$

Let $$Y = (\textbf{a})\cdot x_1 + (\textbf{a}+1)\cdot x_2 + (\textbf{a}+2)\cdot x_3 + (\textbf{a}+3)\cdot x_4 + . . . + (\textbf{a}+n)\cdot x_n$$ Where $x_1,x_2,x_3.... x_n$ are all positive integers ...
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Combinations: There are 30 members

There are $30$ members of the Bay City marching band. Among them, $16$ play the saxophone, $4$ play drums, $8$ play clarinet, and $9$ twirl the baton. No one who plays sax twirls. Everyone who plays ...
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Probability that taxi requiring repair is dispatched to airport C

I'm working through the textbook Mathematical Statistics with Data Analysis 4th edition and didn't understand the solution for question 2.25. It refers to the previous question 2.24 which asks how ...
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Three different medals- gold, silver and bronze- are awarded to athletes in TWO different races…

Three different medals- Gold, Silver and Bronze- are awarded to athletes in two different races. If no athlete may win more than one medal, and there are 6 athletes in total, how many different ...
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$\sum_{k=0}^n (-1)^{k} {{m+1}\choose k }{{m+n-k}\choose m }= 0$ if $n>0$

I need to show that $\sum_{k=0}^n (-1)^{k} {{m+1}\choose k }{{m+n-k}\choose m }= 0$ if $n>0$. Here $m$ is a non negative integer. I am thinking induction, but do I apply it on $m$ or $n$? I ...
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Combinatorics, special permutations of cubic cells.

I'm trying to produce a set of permutations of 3 axis Cartesian coordinates which are limited by a specific geometric constraint: Perhaps the simplest way to visualize this is with a Sudoku-like game ...
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Can any counting situation that works for one side of an identity work for the other (combinatorial proof)

If I come up with a situation that works for one side of a combinatorial proof, does some interpretation always exist for how the other side counts that same situation? Or is it possible that one ...
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Creating a generating function for the Stirling transform

Does there exist a sequence $c_n$ such that $$S(n, k) = \frac{c_n}{c_k c_{n - k}}$$ for $0 \leq k \leq n$, where $S(n, k)$ are the Stirling numbers of the second kind? I ask because I'm trying to ...
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What is the formula for the number of connected graphs with N vertices of max. degree up to 4? Leaving $F(x) = x + x^2 + 2x^3 + 6x^4 + 21x^5 + …$

It is known that F(x) is the generating function of the counting sequence of connected simple graphs with N vertices is given by: $F(x) = x + x^2 + 2x^3 + 6x^4 + 21x^5 + 112x^6 + ...$ where the ...
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With repetition, no order. Probability of drawing atleast 18 distinct balls, when there are 20 different balls and one has 50 draws?

I came up with this today, however I could not figure out a solution to this problem. Say, you have a pool of 20 different balls. Therefore, the probability of drawing one of the balls is equal to ...
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Solution mod 2 of polynomial equation

Can one find a polynomial $p(x,y)$ such that it is integer for integer $x,y$ and it satisfies $$p(x,y) + p(y,x) = x^2 + y^2 +1 \mod 2$$ or prove that it is not possible?
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Probability of drawing exactly X previously unseen numbers

Let's say we have $M$ balls, numbered $1$ to $M$, in a sack. Let us say we have already seen $Y$ of those balls. We now draw $N$ balls without replacement. What is the probability of seeing exactly $X$...
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Permutations/Probability [duplicate]

A club has $n$ members and $r$ officers. In how many ways can we choose $r$ different officers from the members of the club? Answer would have to start with: # of ways to pick $r$ officers from $n$ ...
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