# 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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### Number of binary strings of length $n$ satisfying specific (ad-hoc) conditions

Count the number of binary strings of length $n$ that satisfy the following additional conditions: a) Two zeroes in a row are not allowed b) Three ones in a row are not allowed c) The ...
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### Doubt in finding number of integral solutions

Problem : writing $5$ as a sum of at least $2$ positive integers. Approach : I am trying to find the coefficient of $x^5$ in the expansion of $(x+x^2+x^3\cdots)^2\cdot(1+x+x^2+x^3+\cdots)^3$ . ...
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### Half from any $2n$ but not $2n+2$

Let $n$ be a positive integer. What is the length of the longest possible sequence of $0$'s and $1$'s such that among any $2n$ consecutive numbers, exactly half are $0$'s, but among any $2n+2$ ...
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### Expected Value for Heads for Unknown Weighted Coin Given Head First Flip

This is a combinatorics problem, and I think it involves expected values and conditional probability, but I don't know how to use them: "A bag contains an infinite number of coins whose probabilities ...
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### distributing numbered balls with duplicates into 4 boxes [closed]

How many ways are there to distribute 52 balls, numbered 1 to 13 with 4 duplicates for each number, into 4 distinguishable boxes.
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### Finitely many steps to $n$-stone pile.

I have a combinatoric problem still unsolved: $2n$ ($n$ is a positive integer) stones are divided into $3$ piles. In each step, we pick half of a pile which has even number of stones and move those ...
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### code to calculate all combinations [closed]

I have the follwing problem: I have an array u of length d. The sum of every integer of this array u should be k, and integers are from 0 to k. I want now all possible combination that suffice this ...
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### Bayesian probability problem?

Problem: In a city there are three types of taxis which drive towards the airport. 30% are blue, 20% green, 50% yellow. They take there customers too late with probabilities 0.1,0.2,0.3 respectively. ...
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### Maximum area of triangle inside a convex polygon

Prove that within any convex polygon of area $A$, there exists a triangle with area at least $cA$, where $c=\tfrac{3}{8}$. Are there any better constants $c$? I'm not sure how to approach this ...
### Prove that if a collection of subsets of {1,..,n} that each pair of subsets has at least one element in common, there are at most $2^{n-1}$ subsets
Full question: Prove that if a collection of subsets of {1,2,...,n} has the property that each pair of subsets has at least one element in common, then there are at most $2^{n-1}$ subsets in the ...
The problem is the one on the title. Given a N, find all possibilies for A, B and C that make this true: $AB+BC+AC = N$when $A \ge B \ge C$. This code in C do the job: ...