# 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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### Permutation, Arranging letters [on hold]

Please help me! I am in a hurry! The six letters of the word “MOTHER” are rearranged in all possible orders and the words so formed are listed in alphabetical order
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### Proving an inequality about a set of combinations.

Suppose $A$ is a set of $r$ combinations of an $n$ set, with $\alpha \cap \beta \neq \phi$, whenever $\alpha, \beta \in A$. Show that $$|A| \leq \binom{n-1}{r-1}$$ if $r \leq \frac n2$. What does ...
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### Number of positive unequal integer solutions of $x+y+z+w=20$

What is the number of positive different integer solutions of $x+y+z+w=20$, where $x,y,z,w$ are all different and positive? It would be nice if coding is not used. I am given the answer $552$.
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### Number of permittable numbers given following conditions.

What are total numbers belonging to $\mathbb Q$ (rational) between $2008$ and $2009$ such that after decimal point their digits occur in decreasing order? \begin{align} 1) &\ 9Pi;i\in [1,9], \\ 2)...
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### Number Theory Characterization Problem

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...
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### Prove that if B(0) = 0, then A(B(x)) is a formal power series

I'm working through my Combinatorics textbook and am stuck on this proof. The textbook explains it pretty well, but I am having trouble with one of the steps. I was hoping I could get some help here ...
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### Arrange black and white balls so that each pair of white balls is separated by at least two black balls

I am trying to solve the following question: How many linear arrangements of $m$ white balls and $(n-m)$ black balls are possible such that each pair of white balls is separated by at least two ...
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### Numbers on a circle: how many arc sums can be positive?

There are $n$ real numbers, $a_1,\dots,a_n$, arranged on a circle. Given a fixed integer $k<n$, let $S_i$ be the sum of the $k$ adjacent numbers starting at $a_i$ and counting clockwise, like this (...
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### How many strings of 12 lowercase letters with repetitions

Consider strings of 12 lowercase letters, such as aksdjmnuuyio. How many strings either are a repetition of 2 strings of 6, such as aksdjmaksdjm, or a repetition of three strings of 4, such as ...
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### Optimising my chances of drawing a specific card from a small deck.

I have a deck of 24 cards, 3 of which are aces. I want to figure out my chances of drawing at least one ace based on the number of cards I draw. I'm pretty sure if I draw one card my chance is 12.5% I'...
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### Permutation of coefficient with coditions [closed]

I have 6 coefficients, (V1,V2,H1,H2,D1,D2). Their permutation is 6! = 720. But I have a rule: V2 cannot lead V1, H2 cannot lead H1 and D2 cannot lead D1. For example: V2V1H1H2D1D2 is prohibit. ...
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### How to approximate the Langford numbers with probability?

A Langford pairing, also called a Langford sequence is a permutation of the multi set {$1,1,2,2, \dots, n,n$} in such a way that there are exactly $k$ elements in between every $k$. Interestingly, ...
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### Combinations of 5 integers from 1 to 100 such that differences between the sorted integers of each combination is at least 5 but not more than 10?

For example , I am trying to count combinations like [1,6,14,21,27] because the minimum difference between two sequential integers in the combination is 5 and the maximum distance is 8, but I don't ...
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### Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
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### longest way to rearrange students before returning to original arrangement? [closed]

This is Q24 from the 2012 Intermediate Australian Mathematics Competition: "A teacher has a class of twelve students. She thinks it would be a nice idea if they change desks every day, so she has ...
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### a collection of 20 marbles from infinite pool of 2 color marbles with replacement.. [closed]

I have an infinite supply of pink and blue marbles. Probability that any random draw will yield a pink marble is "p" and prob. of picking blue is 1-p. Let us assume p=0.4 if a numeric value helps. ...
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### Counting non-negative integral solutions

I'm reading this passage and wondering why Number of ways in which k identical balls can be distributed into n distinct boxes = $$\binom {k+n-1}{n-1}$$ could someone explain it to me please?
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### Is there a faster way of computing the probability of a sum $S$ when $n$ dice are rolled? [duplicate]

So far, I've only had to deal with $2$ dice or $3$ dice problems. For example, if the problem asks to find the probability that a sum of $8$ will be achieved from rolling $3$ dice, I just list all the ...
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### number of inversions in permutation if subarray of permutation is reversed?

I have permutation(P) of numbers 1 to N (<=10^5) . Suppose I can reverse the subarray of ...
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### There are 10 boxes, 15 balls; 10 red, 5 blue. Each is randomly placed in a box in an independent manner. What's E[X=the number of empty boxes?]

There are 10 boxes, 15 balls; 10 red, 5 blue. Each is randomly placed in a box in an independent manner. The red balls are placed in boxes 1-10, blue balls are placed in 1-6. What is the expected ...
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### Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
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### How many arrangements of the letters in the word CALIFORNIA have no consecutive letter the same?

First off, the correct answer is $$584,640 = {10!\over 2!2!}- \left[{9! \over 2!}+{9! \over 2!}\right] + 8!$$ which can be found using the inclusion-exclusion principle. My own approach is different ...
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### Prove that a graph has a cycle of length no more than $14$

A graph contains $2016$ vertices, its chromatic number is $5$, prove that this graph has a cycle of length $\leq 14$. Where do I start?
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### The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
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### Why doesn't this alternative method work? Chance of getting four of a kind in a hand of $5$ cards?

Please note: This is not a duplicate since it is asking about an alternative method of solving the question What is the probability of getting four of a kind in a hand of $5$ cards from a standard ...
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### Find all solutions to $2 x + 3 y + 4 z = 10$

I do not have a background in math, and am wondering what type of question this is. I looked combinatorics optimization, and the knapsack problem, but found the vocabulary too dense. The problem: ...