# 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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### A multiple choice question.

A club with $x$ members is organized into four committees such that $(a)$ each member is in exactly two committees, $(b)$ any two committees have exactly one member in common. Then $x$ has $(A)$ ...
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### Solving an Equation involving Factorials and Exponents.

Is it possible to find the value of $n$ in: $$\Large{\frac { (_{ 365 }{ P }_{ n }) }{ { \left( 365 \right) }^{ n } } \quad \approx \quad \frac { 1 }{ 5 }}$$ Please help! Thanks for your answers in ...
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### Seperate the numbers into pairs

With how many ways can we separate the numbers $\{ 1,2,3, \dots, 2n\}$ into $n$ pairs, when: We don't care about the order of the pairs We care about the order of the pairs  At the case when ...
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### With how many ways can we choose cards?

With how many ways can we choose, from usual pack of cards with $52$ cards(that are separated into $4$ colours and $13$ kinds) $5$ cards, $2$ of which should be red($\diamondsuit$ or $\heartsuit$) ...
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### Find out in how many ways the operation can be performed?

i) In how many ways can a committee of $5$ or more be formed from 12 persons? ii) In how many ways can a committee of $5$ be formed from 12 persons if only two of a group of $3$ persons must always ...
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### How many messages in the Morse code?

A message in the Morse code is a finite sequence(a word) with dots, dashes and gaps. How many different messages can be made with $7$ dots,$3$ dashes and $2$ gaps? And how many if it is not allowed ...
200 views

### Telephone Numbers without repeated digits

In a city with telephone numbers with $6$ digits,how many telephone numbers exist without repeated digits?What does this mean? That my telephone number shouldn't have twice the same number or that my ...
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### Counting proof of choosing

I'm doing an exam review without any solutions. I don't know why this is true. $$∑_k^n 5^k\binom{n}{k} = 6^n$$
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### A question on an asymptotic combinatorial expasion

Suppose we are given $(\lambda a + \bar{\lambda}b+O(\lambda^2))^{n}$, where $0 < \lambda < 1$ and $\bar{\lambda} := 1-\lambda$; also, $0 < a,b < 1$. $O(\cdot)$ is the traditional Big-Oh ...
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### How to prove this sequence of inequalities

The number $c_{g}(n)$ is defined by the recurrence $$c_{g}(n) = c_{g}(n-1)+ (n-1)(n-2)c_{g-1}(n-2) ,$$ with $c_{0}(n)=1$ for any $n\geq 1$ and $c_{g}(n)=0$ if $n \leq 2g$. ...
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### partitioning numbers from 1 to n in 4 non-empty subsets so no subset has 2 consecutive numbers.

Attempt: We have to find the number of ways to partition the numbers 1 to n into four non-empty subsets so that none of them are empty. let $f(n)$ be the way to do that and let $f'(n)$ be the way to ...
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### About partitions, majorization, and conjugates [duplicate]

I am trying to solve a question of a property of conjugation. What I am trying to show is that conjugation reverses the order of majorization. Let $\lambda$ and $\mu$ are partitions of n and $\lambda$...
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### In how many ways we can choose $3$ subsets from set $|S| = 20$ …

In how many ways we can choose $S_1$, $S_2$ and $S_3$ from a set which consists of $20$ element, so that : $S_1 \cap S_2 \cap S_3 = \emptyset$
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### chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
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### How to interpret $(2n)!$

It's all in question: how to interpret the factorial from $2n$? Is $(2n)!$ equal to $n!\times n!$ ? The problem is in Combinations if the combinations is $\binom{2n}3$. P.S. The main problem is ...
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### Show the relationship between the trace and the number of 4-cycles

Let $G$ be a k-regular graph. Show the exact relationship between $tr(A^4)$ and the number of 4-cycles in $G$. I understand how $tr(A^4)$ tells us the total number of closed paths of length 4 in $G$,...
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### Number of ways to add up to a number without repetition (order does not matter)?

I have a number x and want to find how many ways there are to add up to that number using the y numbers from numbers 1-z. for example, for x=15 y=3, z=9, there are 8 ways to add up to 15 using 3 ...
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### Calculation a closed form for the sum

Please help me to calculate this sum in a closed form: $$\sum\limits_{1\leq i_1<i_2<\ldots<i_k\leq n}(i_1+i_2+\ldots+i_k).$$ Here $n$, $k$ are positive integer numbers; $k<n$. I think ...
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### choosing poker hand with a specific card

How many ways can you choose at least one A from a deck of card in a poker hand? I just wanted to double check my answer, would it be C(52,5)- C(48,5) Help is much appreciated,
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### Different ways of picking a committee of $12$ women and $10$ men

$12$ women and $10$ men are on the faculty. How many ways are there to pick a committee of $7$ if (a) Claire and Bob will not serve together, (b) at least one woman must be chosen I'm not sure ...
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### How to calculate the following sums?

I would like to know of a way to evaluate the following two for arbitrary $n$. $$\sum_{i=1}^ni!\,, \quad \sum_{i=1}^n \frac{n!}{i!}.$$
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### rolling dice 6 times, outcomes showing of 2 sixes

If 6 dices are rolled, in how many ways will exactly 2 sixes show up? I was thinking that it would be 6*6*5*5*5*5, am I right?