# 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.

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### Number of angarams of BORBOTTIO in which same letters are close together

I had the following problem: We have the word BORBOTTIO. Find: All the anagrams All the anagrams that starts with BB All the anagrams where the same letters are close together. The first two ...
1answer
26 views

### expectation of number of hands shaken at round table

There are $x$ people from country A and $y$ people from country B. They sit around a table and shake hands with people on their left and right, but they only shake hands if they're from the same ...
1answer
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### Even coin tosses, 2 players

The game consists of a sequence of independent plays for each of the two players. Each round I have probability $p$ of making a point and my opponent has probability $1-p$ of making a point. (so each ...
0answers
32 views

### A conjecture on tiling

Here's the problem: An $L-tile$ is the one which looks like this, and covers $3$ square units: A A A Now we will define a term called an $L_k-good$ rectangles where $k$ is a nonnegative ...
0answers
16 views

### Minimum difference between two subsets

This problem feels very basic, however I could neither google it nor solve it satisfactorily: Given a set of positive numbers $X \subset \mathbb{R^+}$, find the minimal nonzero difference between two ...
0answers
18 views

### Why is the “q-Pochhammer Symbol” referred to as such, despite not being a q-analog of the Pochhammer symbol?

The q-Pochhammer symbol (for $k>0$): $(a;q)_k = \prod\limits_{j=0}^{k-1}\left(1-aq^j\right)$ The Pochhammer symbol / rising factorial: $(a)_k = \prod\limits_{j=0}^{k-1}(a+j)$ The falling ...
1answer
19 views

### Finding the general recursive formula to divide the pot fairly

In a game where each turn is made up of the roll of a die, player E gets one point when the die is even, and player O gets one point when the die is odd. The first player to accumulate 7 points wins ...
1answer
21 views

### Finding a good bound for a quadratic and exponential expression of integers

Suppose you have two collections of $k$ positive integers, $a_{1},\ldots,a_{k} \in \mathbb{N}$ and $b_{1},\ldots,b_{k} \in \mathbb{N}$. Suppose you have bounds on the sum $\sum_{i=1}^{k} a_{i} \leq A$...
0answers
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### Smallest number not expressible using all of the first $n$ powers of $2$ (once each), with $+$, $-$, $\times$, $\div$, and parentheses?

Motivation Solution to this problem is a lower bound for a more general problem. Problem Given first $n$ powers of two: $1,2,4,8,16,\dots,2^{n-1}$ that all need to be used exactly once per number ...
1answer
22 views

### How many ID numbers exist? - check my solution

"An ID number is created as follows: The first part must consist of at least one letter and at most of four letters (the alphabet is assumed to have 26 letters). The second part must consist of at ...
0answers
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1answer
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### Counting the number of ways to divide into teams - complicated

$n$ students are standing in a row. Teacher must divide them into smaller teams - it could be one team or more - (team must consist of students standing next to each other in a row) and choose in ...
4answers
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### Combinatorics problem: How many ways this row can be filled?

A row measuring $N$ units in length has $M$ red blocks with a length of one unit placed on it, such that any two red blocks are separated by at least one grey square. How many ways are there to ...
0answers
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### selecting numbers from the group [on hold]

What is the number of options for selecting 80 numbers from the group {10.11 .... 99} with repetitions and no matter how orderly, each number that divides by 10 without residual is taken at least ...
0answers
15 views

### Bijective mapping to fixed image set

I need to reduce the amount of stored data in an Ethereum Smart Contract. Currently I am storing $n < 2^8$ strings in a mapping (i.e. a map data structure) and each string is mapped into $[0, n)$; ...
2answers
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0answers
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### Asymptotic Gilbert-Varshamov Bound Using Hilbert's Entropy Formula

I am reading Walker's book Codes and Curves and am having trouble proving this Lemma regarding the Asymptotic Gilbert-Varshamov bound. Suppose that $q$ is a prime power and we define \begin{align*} ...
1answer
59 views

### How many bit strings of length 20 have exactly five $1$’s and do not contain $11111$ as a substring? [on hold]

Recall that a bit string is a string composed of characters $0$ and $1$. Can someone explain how the answer is: ${20\choose5} - 16$?