The inclusion-exclusion principle states that the number of elements in the union of two given sets is the sum of the number of elements in each set, minus the number of elements that are in both sets.

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In how many ways can $1000000$ be expressed as a product of five distinct positive integers?

I'm trying to solve the following problem: "In how many ways can the number $1000000$ be expressed as a product of five distinct positive integers?" Here is my attempt: Since $1000000 = 2^6 \cdot ...
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Inclusion Exclusion with 4 sets: How many integers between 1 and 100 are divisible by 2 or 3 or 5 or 7?

How many numbers between 1 and 100 are divisible by 2 or 3 or 5 or 7? The solution I had gives a different answer from what was provided, so I was wandering if anyone could tell me what mistake I ...
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inclusion–exclusion principle formula in probability for general case

For the system reliability analysis of complex systems, I'll use the formula below after a point in a custom computer code. I know that for the n=2 case, formula result is $P_1+P_2-P_1.P_2$ ($P$ : ...
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Prove the inclusion-exclusion formula

We just touched upon the inclusion-exclusion formula and I am confused on how to prove this: $|A ∪ B ∪ C| =|A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|$ We are given this hint: To do ...
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Number of permutations of $A=\{1,2,3,\dots,n\}$ such that $|x_i-x_j|\ne|i-j|$ of every $i,j\in A$

Let $B$ be the permutation of $A=\{1,2,3,\dots,n\}$ such that $|x_i-x_j|\ne|i-j|$ of every $i,j\in A$ where $x_k$ is $k-th$ element of $B$. How many different $B$ exist? On first sight it doesn't ...
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Inviting 4 friends out of 8 for a week such that each friend visits at least once

Dave is inviting 4 friends out of 8 for a week how many possibilities there are such that each friend visit at least once. Let's number the friends for brevity, 1 to 8. This is like asking how ...
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31 views

Probability of placement in a random ordering of integers (inclusion-exclusion rule)

Problem Let $a_1, ..., a_n$ be a random ordering of integers $1,...,n$. What is the probability that there exists no $1 \leq i \leq n$ such that $a_i = i$? Attempted solution For #1: let $A_i = ...
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22 views

Summation with Multiple Indices

How would the following summation work? $\sum_{r,s,t \ge 0_{r+s+t=n}} \binom{m_1}{r} \binom{m_2}{s} \binom{m_3}{t}$ How would you choose the value for the next integer in the series? For example, ...
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64 views

Inclusion-Exclusion question

A careless payroll clerk is placing employees’ paychecks into pre-labeled envelopes. The envelopes are sealed before the clerk realizes he didn’t match the names on the paychecks with the names on the ...
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Determine the number of ways for n couples to stand in a line so that no one stands beside his or partner (explanation for the answer)

I'm not quite sure if I'm understanding solution to following problem: "Determine the number of ways for n couples to stand in a line so that no one stands beside his or her partner." The general ...
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number of pairs formed from $2n$ people sitting in a circle

I am trying to understand the solution to the following problem: Suppose that $2n$ persons are sitting in a circle. In how many ways can they form $n$ pairs if no two adjacent persons can form a ...
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42 views

Question about probability theory inclusion/exclusion

So, I've been trying to teach myself probability theory, and I think I have the idea down, but I wanted to make sure I am on the right track. Thus, I was wondering if people here could confirm that I ...
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104 views

There are 5 students. Must form 3 teams and every team must have 2 students in common.

This is the problem. Suppose there are 5 students and we are trying to form 3 distinct scientific teams such that Every student must join into at least one team (no student should left without a ...
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58 views

Ways to distribute candies using inclusion/exclusion

I have 173 million pieces of candy. There are 5 students. In how many ways can this distribution be made, assuming that each student receives at least 1 million candies, Student B receives at most 10 ...
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60 views

How to extend the Inclusion Exclusion Principle to an XOR situation

For calculating the probability of the union of independent events, $P(A_1\cup A_2\cup A_3\cup A_4)$ one can use the Inclusion Exclusion principle: $$ \eqalign{ P\Bigl(\bigcup_{i=1}^n A_i\Bigr) = ...
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28 views

Number of ordered quadruples sets

I'm trying to tackle the following question, but got no clue how to do so (I know it is not very popular not showing any effort, but this time I really don't know what to do). Iv'e seen a solution ...
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If we are given that a list of $n$ numbers has $11,660$ derangements, what is the value of $n$?

The Full Question For the positive integers $1,2,3,\dots n-1,n$, there are $11,660$ where $1,2,3,4,5$ appear in the first five positions. What is the value of $n$? My Work First I considered all ...
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Simple combinatorics - inclusion and exclusion problem

There are 3 groups ${{A}_{1}},{{A}_{2}},{{A}_{3}}$ What we know about them : \begin{align} & |{{A}_{1}}|=|{{A}_{2}}|=|{{A}_{3}}|=n \\ & |{{A}_{1}}\cup {{A}_{3}}|=2n-q \\ & ...
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23 views

# of Numbers from 1 to 2400 which are divisible by (8 or 5) or not by 6

I need to apply the Principle of Inclusion-Exclusion here: Hence I get $$|A_8| + |A_5| + |\neg A_6| - |A_8 \cap A_5| - |A_8 \cap \neg A_6| - |\neg A_6 \cap A_5| + |A_8 \cap A_5 \cap \neg A_6|$$ I ...
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69 views

Understanding inclusion exclusion principle (with example question).

Hi here is a question that i solved with generating functions , and i try to solve the same question with the inclusion exclusion principle. Question: We have four type of balls - ...
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100 views

let s be a set with N elements and A1,…,A101 be 101 (possibly not disjoint) subsets of S

So the question I'm having problem with is the following: let s be a set with N elements and A1,...,A101 be 101 (possibly not disjoint) subsets of S with the following 5 properties: each elements ...
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99 views

How many lattice paths with step S and W are there that begin at (0,0), end at (-12,-12)

How many lattice paths with step $S$ and $W$ are there that begin at $(0,0)$, end at $(-12,-12)$ and do not go through any of the points $(-1,-4), \space (-5,-3), \space (-9,-11)$? I'm unsure of how ...
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Distribute $10$ distinct prizes to $4$ students

Inclusion Exclusion Used Theorem 8.1 The Question: In how many ways can one distribute $10$ distinct prizes to $4$ students with exactly $2$ students getting nothing. b) How many ways have at ...
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75 views

How Many Ways Can 3 x's 3 y's and 3 z's be arranged so that no consecutive triple of the same letter appears

My Question I want to know two things: is my solution correct, and is there another, more clever, way to solve it? My Solution There are a total of $\frac{9!}{3!3!3!}$ arrangements of $xxxyyyzzz$. ...
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Seating permutations for 10 people where 2 people always sit together and 2 people never sit together

We have to seat 10 people in a row. Condition: two people always sit together and two people never sit together. My attempt: Let the two people who always sit together be taken as 1 person for the ...
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44 views

Number of lists of given size with given max element value where K cells (possibly overlapping) can have only multiples of certain numbers

I have been trying to wrap my head around a problem. The problem is reduced form of a problem from a programming contest. I have been trying to apply inclusion-exclusion principle toward solving this, ...
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Can the principle of inclusion/exclusion be used to count elements in the intersection of a sequence of sets?

The principle of inclusion-exclusion (PIE) is often used to count the number of elements in a union of $n$ sets in terms of an alternating sum of their various intersections: $$ \left |\bigcup_{i \in ...
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Use the PIE to prove an identity

Use the PIE to prove, $n!{{n-1}\choose {k-1}} = \sum_{i=1}^k (-1)^{k-i} {k \choose i} \bar i^{n}, \ 1 \le k \le n$ Where, $\bar i^{n}= i(i+1)...(i+n-1)$ Edit Actually,I can solve it now. And it's ...
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Solving probability question using the inclusion exclusion principle

Question: $F$ is the set of all functions from a $n$-set to $B=\{a,b,c\}$. In a random experiment of selection of functions from $f$ assume that every $f\in F$ is equally likely. What is the ...
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21 views

No. of ways to shuffle a card

How many ways are there to shuffle N cards such that exactly one card is in the same position?(Assuming that initially the card no. 1 is in the first position,card no.2 is in the second position and ...
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88 views

Probability of $m$ out of $n$ rolls of a die being among the numbers $1,2,\ldots,m-1$, for some $m$.

Suppose I have a $k$ sided die with the numbers $1,2,\ldots,k$ on each side, and that I roll it $n$ times ($n<k$). What is the probability that there exists an $m\leq n$, so that $m$ of the $n$ ...
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Inclusion and Exclusion Principles

In a scientific study of 233 imaginary people, each eats at least one meal every day. Of these, 91 eat breakfast, 152 eat lunch and 177 eat dinner. Also, 190 eat either breakfast or 1 lunch, 205 eat ...
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Principle of inclusion and exclusion

How many integer solutions can we have for the equation $x_1 + x_2 + x_3 = 18$, if $0 \leq x_1 \leq 6$, $4 \leq x_2 \leq 9$ and $7 \leq x_3 \leq 14$? Using the Principle of inclusion and Exclusion: I ...
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Finding the range of a function given the cardinality of two sets

If I am given that a set $A$ has size $5$ and another set $B$ has size $4$, how many functions of range with cardinality $3$ can there from $A$ to $B$? I am unsure as to how to go about solving this ...
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Solving a question about inclusion exclusion principle

I am trying to solve the following question: There are $50$ students in a class who are given a test with $3$ questions on it: $Q_1$, $Q_2$, and $Q_3$. All the students answer at least $1$ question. ...
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How many words can be written with $aabbbccdd$ such that no two equal letters are adjacent?

I'm trying to count this using the principle if inclusion-exclusion. I've done the following: Counting the number of permutations of $aabbbccdd$. $9!$ Counting the number of ...
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21 views

compute the number of permutations

Compute the number of permutations of $\{1,2,3,4,5,6,7,8,9\}$ in which either $2,3,4$ are consecutive or $4,5$ are consecutive or $8,9,2$ are consecutive. I know we will use some exclusion-inclusion ...
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118 views

Prove combinatorial identity using inclusion/exclusion principle

The identity is: $$\sum_{k=0}^{m} (-1)^{k} {{n} \choose {k}}{{n-k}\choose{m-k}} = 0$$ I'm not even sure where to begin. Does anyone have any suggestions?
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50 views

Permutation (inclusion-exclusion)

2 corrected exams are being returned to each of n students. How many ways can the teacher give those 2 exams back to each student such that everyone receives at least 1 exam that is not his. I know ...
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Exclusion-inclusion principle- task.

How many such numbers in 5-digit decimal expansion such that: (1) 3rd digit is 7 and digit 5 there is neither time or (2) there is no number more than once. My solution: Let a set $A$ is a set of ...
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32 views

Enumeration of integers are in increasing order which have gaps

I want to solve the following: Calculate the number of ways of selecting five distinct integers $x_1,x_2,x_3,x_4,x_5$ where $0\leq x_1 \lt x_2 \lt x_3 \lt x_4 \lt x_5 \leq 20$ I think this may ...
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Using inclusion-exclusion principle to count the integers in $\{1, 2, 3, \dots , 100\}$ that are not divisible by $2$, $3$ or $5$

Question: How many integers in $\{1, 2, 3, \dots , 100\}$ are not divisible by $2$, $3$ or $5$? Can anyone give me a full explain of how this applies to the inclusion-exclusin principle? Because I ...
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Solution for equation with inclusion-exclusion principle.

Using the principle of inclusions and exclusions count how many solutions to the equation $$ x + y + z = 12 $$ $$ 1 \le x \le 5$$ $$ -2 \le y \le 4$$ $$ 0 \le z \le 5 $$ $$ x,y,z \in \mathbb{Z}$$
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Number of simple graphs with no vertices of degree 0

Determine the number of graphs with no vertices of degree 0 on a given $n$-element vertex set V. The total number of simple graphs with $n$ vertices is $2^{\binom{n}{2}}$. We want to find the number ...
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39 views

how to calculate these intersections without having to count all combinations

We have the following sets: $X= {(a,b,c,d) ∈S: b< c < d},$ $Y= {(a,b,c,d) ∈S: a< c < d},$ $Z= {(a,b,c,d) ∈S: a< b < d},$ $F= {(a,b,c,d) ∈S: a< b < c},$ Where each of ...
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proving an invloved combinatorial identity

How to prove following Identity? $$\sum_{k=0}^n (-1)^k {n-k \choose k} m^k (m+1)^{n-2k} = \frac {m^{n+1}-1}{m-1}, m \ge 2$$ This seems very hard to me. Any idea about how to prove it combinatorialy? ...
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31 views

Task using principle of inclusions and exclusions.

Let $A = \{ 1,.....,100000\}$. Using principle of inclusions and exclusions determine cardinality $B$. Let $B$ be such subset $A$ that $e \in B \iff e $ is a number which contains at least one digit ...
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115 views

Applying the inclusion exclusion principle to count permutations with forbidden subsequences

I have problem with this: How many permutations of the letters A,B,D,E,I,K,M,N,R,U,Z are there so that none of the words: ARZEN, DRAK, DUM, DURAZ are subsequences of the permutation. This means it ...
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1answer
116 views

Prime number in set $\{1,…,60\}$

How can we calculate by using the principle of inclusions and exclusions how many prime numbers are in the set $ \{1, ..., 60 \} $?
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27 views

Calculating number of students who don't study any language

According to a survey of 100 students, there are 40 students studying English, 30 studying French, and 25 studying Spanish. Inaddition, 8 students are studying English and French, 6 are ...