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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.

Combinatorics is the study of finite or countable discrete structures — especially enumerative combinatorics: how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. This tag can be used for questions about permutations, combinations, partially ordered sets, bijection proofs, and generating functions.

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To learn more about enumerative combinatorics, see Wikipedia or Richard P. Stanley's Enumerative Combinatorics.

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How to distinguish between combination and permutation questions?

How do you distinguish combination and permutation question? An example of a combination question: Example: How many different committees of 4 students can be chosen from a group of 15? Answer: There are possible combinations of 4 students…
guest11
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A set contains $\{1,2,3,4,5....n\}$ where $n$ is a even number. how many subsets that contain only even numbers are there$?$

A set contains $\{1,2,3,4,5....n\}$ where $n$ is a even number. how many subsets that contain only even numbers are there for the set$?$ This is my solution, is this valid$?$ since number of single element subset that contain only a even number is:…
kokoro
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Proving that $\sum_{a=1}^{b} \frac{a \cdot a! \cdot \binom{b}{a}}{b^a} = b$

Prove that for all positive integers $b$ that $$\sum_{a=1}^{b} \frac{a \cdot a! \cdot \binom{b}{a}}{b^a} = b.$$ My idea is induction, but I cannot figure stuff out on the inductive step.
user284139
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Proof of subfactorial formula $!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$

Any hints about how to prove $$!n = n!- \sum_{i=1}^{n} {{n} \choose {i}}\,!(n-i)$$ from Wikipedia's article on derangements? Here, $!n$ is the number of derangements of a set with $n$ elements. I am not looking for proofs, just nudges in the right…
picakhu
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Words of Length $n$ over the Alphabet $\{1,2,3\}$ with Certain Restrictions

Let $w(n)$ denote the number of words of length $n$ over the alphabet $\{1,2,3\}$ with the restrictions that in a word the parity of $1$s be even and the parity of $2$s odd. I have written out the possible words for some small values of $n$, hoping…
Holdsworth88
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Simplify $\sum \limits_{k=0}^{n} \binom{n}{k} 2^{\sqrt{k}}$

Can this sum be simplified: $\sum \limits_{k=0}^{n} \binom{n}{k} 2^{\sqrt{k}}$ Or at least is there a simple fairly tight upperbound? EDIT So I think this sum is more easily bounded than I previously thought: Clearly, $$\sum_{k=0}^{n} \binom{n}{k}…
Travis Service
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Different shapes made from particular number of squares

Good day! I’m currently investigating how different shapes can be made from a particular number of squares. I have two major concerns: (1) Will there be a formula predicting the number of shapes that can be made from a certain number of squares…
wildberry tart
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Minimal number of animals in a matching card game

I saw a card game designed for small children. Each card has a picture of 6 animals on it, and there are 31 cards. When any two cards are compared to each other, they share exactly one animal. The game is for the child to find the animal that is…
Brandon Yarbrough
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How many different dice exist? That is, how many ways can you make distinct dice that cannot be rotated to show they are the same?

Dice are cubes with pips (small dots) on their sides, representing numbers 1 through 6. Two dice are considered the same if they can be rotated and placed in such a way that they present matching numbers on the top, bottom, left, right,…
Connor
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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*} \zeta(2,1)=\zeta(3)=8\zeta(\overline{2},1) \end{align*} Such…
epi163sqrt
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Number of non-decreasing functions?

Let $A = \{1,2,3,\dots,10\}$ and $B = \{1,2,3,\dots,20\}$. Find the number of non-decreasing functions from $A$ to $B$. What I tried: Number of non-decreasing functions = (Total functions) - (Number of decreasing functions) Total functions are…
Sudhanshu
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Sum of 1.5-powers of natural numbers

I recently have met the following approximate equation: $$\sum_{k=1}^n k^{1.5}\approx\frac{n^{2.5}+(n+1)^{2.5}}{5}.$$ It's a rather accurate approximation (for $n=40$ the absolute error is $\approx 1.67$ and it increases very slowly), and looks…
Joker_vD
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How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that post but without success. Any idea?
No_way
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(Fast way to) Get a combination given its position in (reverse-)lexicographic order

This question is the inverse of the Fast way to get a position of combination (without repetitions). Given all $\left(\!\!\binom{n}{k}\!\!\right)$ combinations without repetitions (in either lexicographic or reverse-lexicographic order), what would…
van
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Why isn't there only one way of painting these horses?

If you have $11$ identical horses in how many ways can you paint 5 of them red 3 of them blue and 3 brown? My intuition instantly tells me there is only one way of doing this. I mean if the horses were distinct I know there would be…
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