# 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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### Total number of perfect square which are factors of n [closed]

A number $N$ can be factorized as $$N = p_1^5 p_2^4 p_3^7.$$ Find total number of perfect square, which are factors of $N$.
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### Counting spanning trees in labelled graphs

I have some troubles with counting spanning trees, it seems completely abstract to me. First one is cycle with $n$ vertices - it's just $n$, because we can move each number $n$ times like so: $1234$ ...
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### On the GCD of two palindromes.

I had an observation. Which I will discuss below. My question will be Is my observation correct? If so, how can one prove it? Observation: Consider the string of palindromes below: $100...01$ and ...
145 views

### What is the probability of sinking all ships in a simplified game of battleship?

Consider a simplified game of battleship. We are given a $4\times 4$ board on which we can place $2$ pieces. One destroyer which is a $1 \times 2$ squares and a submarine that is $1 \times 3$ ...
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### Finding a ratio from a set of discrete values

For $x = p/q$, where $x$ is a known value between $0.000$ and $1.000$ rounded to the thousandths place, $p$ is an integer value between $0$ and $127$, and $q$ is an integer value between $0$ and ...
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### How many colours do we at least need so that we can ensure all 250 countries have different flags.

One for FN standardized flag consists of three horizontal rectangular fields. If we assume that the middle field not are allowed to have the same colour as the top or bottom field, how many colours do ...
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### How can I prove this equation holds?

As the final part of a big proof I got for uni homework: (It is an extra question, may be unsolvable) $$k^n<\sum_{i=0}^n\binom{n}ik^{n-i}(2^i-1)$$ My idea is to develop the right side into an ...
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### Letter combinatorics and probabilities

Hello I've got some problems and I don't know if my solutions are correct: Given a Text with two letters $A$ and $B$ and the the probability of occurrence of letter $A$ is $p_a$ and $B$ is $p_b$, the ...
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### Sum of odd integers $= x$

How many sums are there that add up to a whole number $x$, and are made of only odd numbers? Each number can be used more than once.
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### Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
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### Tokens in boxes problem

Tokens numbered $1,2,3...$ are placed in turn in a number of boxes. A token cannot be placed in a box if it is the sum of two other tokens already placed inside that box. How far can you reach for a ...
31 views

### Vandermonde-type convolution with geometric term

Is there a closed-form solution to the following sum? \begin{align*} f(r, s, n) = \sum_{k=0}^{n}c^k\binom{r}{k}\binom{s}{n-k} \end{align*} I know this corresponds to find the coefficient of $x^n$ of ...
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### Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$

Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$. Relevant definitions: $G$ is a simple graph that consists of a vertex set ...
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### Probability - Combinations

I am having big problems with this exercise: There are $n$ customers and $k$ types of products and number $i$, where $n \ge k \ge i$. I have to find the probability of the situation where ...
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### Pigeonhole Principle by using induction

Prove the generalized Pigeonhole Principle: Let $n$ and $m$ be natural numbers, $X$ and $Y$ sets with $|X| = mn + 1,\; |Y | = n$, and $f : X\to Y$ a function. Then there exists $y \in Y$ such that ...
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### Number of additive partitions [closed]

Show that the number of additive partitions of $n$ in which no summand appears more than $d$ times equals the number of additive partitions of $n$ in which no summand is a multiple of $d+1$. Now ...
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### Number of possible arrangements of rings on a hand

This is a homework question that I'm having trouble figuring out how to start. Here's the question. A woman has 3 different rings. On any given day she wears 1, 2, or (inclusive) 3 of her rings on ...
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### 12 books shelf and bag.

I got two varieties for the same question: Ways that four books out of a bag of 12 books can be placed on a shelf. Ways to choose 4 books out of 12 arranged on a shelf and put them in a bag. ...
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### Game of Nim: Losing Positions [closed]

If you have heard of the game Nim, this is a version of the game. However, in this version, the players can only remove the amount of stones from the pile which is coprime to the current pile size. ...
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### Asymptotic for combinatorial function

Let $$F_q(k) = \sum_{n=1}^{\infty} \binom{n-1}{k} \binom{1/2}{n} q^n$$ be a function on $\mathbb{N}$. I am interested in the asymptotic behavior of $F$. Any ideas how to tackle it?
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### Arrangements with no anomalous neighborhoods

How many ways can $8$ boys and $20$ girls be ordered such that for each boy at position $i$, there is no neighborhood (of $2n+1$ points with $n > 0$) consisting of positions $j \in [i-n,i+n]$ that ...
18 views

### Suppose a bookshelf contains five discrete math texts, two data structures texts, six calculus texts, and three Java texts

(a) How many ways can you choose one of the texts? (b) How many ways can you choose one of each type of text? Solution: a) By the rule of sum, there are all together $5 + 2 + 6 + 3 = 16$ ...
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### Analysis of sorting Algorithm with probably wrong comparator?

It is an interesting question from an Interview, I failed it. An array has $n$ different elements $[A_1 , A_2, ..., A_n]$ （random order）. We have a comparator $C$, but it has a probability $p$ to ...
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### There are how many ways can we list, without repetition of all the elements of $S = \{ x, y, z\}$

Solution: there are six ways: $xyz$, $xzy$, $yxz$, $yzx$, $zxy$ and $zyx$. Doubt: How do we know there are six possible ways?
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### Seating children in the cinema

I just had finished my class and have been struggling with a problem. There's $9$ seats in the cinema, and two families $F_a=\{F_1,F_2,F_3,F_4,F_5\},$ $F_b=\{F_a,F_b,F_c,F_d\}$ In how many ways can ...
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### How many coefficients in $(x_1 +x_2 + \cdots + x_L)^N$?

How many coefficients in $(x_1 + x_2 + \cdots + x_L)^N$? That is to say, what is the number of coefficients when it represents as sum of products.
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### Birthday problem: why is this solution wrong?

This question is about the birthday problem: the probability that in a group of n people, at least two of them have the same birthday (https://en.wikipedia.org/wiki/Birthday_problem). An easy way to ...
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### How many words can be formed, given $4$ letters, and in each word there must be at least two letters are the same?

How many words can be formed, given $4(a,b,c,d)$ letters, and in each word from $4$ letters there must be at least two letters are the same? The position of the letter doesn't matter. The answer is ...
842 views

### Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

Consider the following identity: $$\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$$ Consider the set $S$ of size $2n-2$. We partition $S$ into two sets $A$ ...
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### inding all possible non-repeating numbers with given digits

How to find all non-repeating number from the following digits:$0,2,4,5,7,8$ This is how I tried to solve it: Since numbers can't start with 0, and the order of the elements matters, it has to be ...
409 views

### Calculating nCr mod M using inverse multiplicative factors

The method used for calculating nCr mod M is: fact[n] = n * fact[n-1] % M ifact[n] = modular_inverse(n) * ifact[n-1] % M And then nCr is calculated as ...
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### In how many ways can $2t+1$ identical balls be placed in $3$ boxes so that any two boxes together will contain more balls than the third?

In how many ways can $2t+1$ identical balls be placed in $3$ boxes so that any two boxes together will contain more balls than the third? I think we have to use multinomial theorem, but I cannot ...
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### Three Digit Numbers Above $560$ Formed From $3,4,5,6,7$

Is there a straight forward way of calculating the number of three digit numbers greater than 560 that can be formed from the numbers $3,4,5,6$, and $7$. I found it to be $30$ but I did it in a round ...
259 views

### Probability of having 'k' elements that occur only once in a multiset filled by sampling with replacement

Let's say that I have a set of unique elements, $P$, and a multiset $M$ that I fill with $N \leq ||P||$ elements by sampling with replacement from $P$. What is the probability that the multiset $M$ ...
523 views

### How many rectangles?

Q: How many rectangles? What should I do here? I don't even know where to start from. Please help me by giving me a hint.
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### IMO Longlist 1989 (Number of ways product can be expressed)

Given two distinct numbers $b_1$ and $b_2$, their product can be formed in two ways: $b_1 \times b_2$ and $b_2 \times b_1.$ Given three distinct numbers, $b_1, b_2, b_3,$ their product can be ...
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