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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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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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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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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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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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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Calculating surjections using inclusion exclusion

I'm trying to find a solution to this problem Find the number of surjective functions from $ \left \{1,2,3,4,5\right \}$ to $ \left \{1,2,3\right \}$ using inclusion-exclusion. I understand what ...
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26 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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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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Easy task to thinking but hard to solve with exclusion-inclusion principle.

Using exclusion-inclusion principle count how many is solutions for: $$x + y + z = 12 \\ x,z \in \{ 3,4,5\} \\ y\in \{2,3,4\}$$ On the whole the task is easy but with using this principle makes it ...
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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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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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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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92 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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Counting Number of Possibilities using Inclusion-Exclusion

I have been tasked with answering the following combinatorics problem for a homework assignment: Consider the set of all six digit numbers that don’t begin with 0. How many of these have at least one ...
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Probability Question, Possible Inclusion/Exclusion?

The question is as follows: Given $x + y$ students in a class, and $r + s$ girls in the class, $x \geq r$. Randomly selecting $x$ students, what is the probability that exactly $r$ of the students ...
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How many balls can be selected?

A sack contain 20 identical red balls, 20 identical blue balls, and 20 identical green balls. In how many distinct ways can 10 balls be selected from the sack?
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Explaining the coefficients of matching polynomials

Matching polynomials are generating functions that tells us the number of $k$-matching (meaning choosing of $k$ independent/non-adjacent edges) in the graph say $G$. Farrell et al., "On matching ...
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Inclusion-Exclusion Principle application Qn

I am suppose to find the number of integers from 1 to 1000 inclusive which are multiples of 3 or 7 or 12: the answer i got was 414 however, the answer stated on the answer sheet provided is 428. Can ...
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counting bitstrings of specific length

Is my solution right refarding this question? How many bitstrings of length 77 are there that start with 010 (i.e, have 010 at position 1, 2, and 3) or have 101 at positions 2,3, and 4, or have 010 ...
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Using Inclusion/Exclusion to solve $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$

I want to solve for: Number of integers solutions to the equation $x_1+x_2+x_3=15$ with $x_1,x_2\leq 5$ and $x_3\leq 7$ for non negative integers $x_1,x_2,x_3$ How can this be done using the ...
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Algebra of sets application

In the problem below algebra of sets is being evaluated, Venn diagrams are allowed. There's a Modern Languages reading exam where 200 students are being evaluated. The exam content is in French, ...
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derivation of derangement with inclusion -exclusion

I read many articles on the derivation of derangement formula but I can't understand them clearly.At first I read articles in the link http://en.wikipedia.org/wiki/Derangement.I understand recursive ...
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A probability problem I am stuck on

A cell contains $N$ chromosomes, between any two of which an interchange of parts may occur. If $r$ interchanges occur then what is the probability that exactly $m$ chromosomes were involved? The ...
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inclusion-exclusion problem

How many base-k sequences are there of length n which contain all k possible symbols? I am trying to solve it using inclusion-exclusion. However, I am actually struggling with the part of counting the ...
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41 views

How many strings are there (inclusion exclusion principle)

Q: What is the number of strings with the length of $8$ above $\left\{1,2,\cdots,10\right\}$ where $7,8$ appears at least one time? So by using the inclusion exclusion principle: $10^8$ ...
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No of distinct combinations of non distinct elements

Example: Given that there are $5$ places that has to be filled with distinct values,, they come from $5$ different bags that contains distinct elements viz $\{1,2,3,5\} ...
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108 views

Number of ways possible to form a number?

Suppose we need to form a 4 digit number with the restriction that ...
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Finding number of combinations for similar items

I have 4 types of popsicle colors: Red, white, yellow, green. In the grocery store, there's 11 popsicles of each type, all together 44 popsicles. I need to find the number of combinations for choosing ...
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Counting problem involving candy and overcounting

I have $40$ chocolates, $29$ gumdrops, and $38$ lollipops. Why is it that the total number of ways to select $41$ candies from this set is: $$\binom{41+3-1}{3-1} - \binom{0 + 3 - 1}{3 - 1} - ...
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Are there further transformation principles similar to the Inclusion-Exclusion Principle (IEP)?

This question is motivated by the elaboration of the question Combinatorial Proof of Inclusion-Exclusion Principle (IEP). Let's consider the following two aspects: 1.) IEP transforms at least ...
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Combinatorics: help with the Inclusion-exclusion principle

$A=\{1,2,3,4\}$ $B=\{5,6,7,8,9\}$ $K$ is a relation from $A$ to $B$ ($K\subseteq AXB)$ in how many $K$'s - $1 \notin domain(K)$ ? in how many $K$'s - $\{1,2,3\}\subseteq domain(K)$ (use the ...
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81 views

Lower bound on union of finite sets (inclusion-exclusion principle)

I have the following problem emerged. Let's say we have $l$ finite sets $A_1, A_2, \ldots, A_l$ with cardinality of $n_1, n_2, \ldots\, n_l$, respectively. We know that $| A_i \cap A_j | \le a_{ij}$ ...
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Deriving the formula for derangements: $\text{Round}\left[\frac{n!}{e}\right]$ [duplicate]

I saw on wikipedia that a formula for derangements is $\text{Round}\left[\frac{n!}{e}\right]$ However, how did they arrive at this elegant formula? Does it have to do with $ !n=n! \sum _{k=0}^n ...
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MathCounts 1993 National Sprint #28

Here is the problem: In a small town of 100 men, 85 are married, 70 have a telephone, 75 own a car, and 80 own their own home. On this basis, what is the smallest possible number of men who are ...
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List 4 partitions of the alphabet containing 3 sets. How many different partitions can be made?

The first part of the question is easy; there are many right answers but I put down: $$(a), (b), (c..z)$$ $$(a,b), (c), (d..z)$$ $$(a,b,c), (d), (e..z)$$ $$(a,b,c,d), (e), (f..z)$$ The part I am not ...
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# of counts of an element - Combinatorial Proof of Inclusion-Exclusion Principle (IEP) [Ross P31]

Let $A_{1},\ A_{2}$, , $A_{n}$ be $n$ sets. Then $|A_{1}\cup A_{2}\cup ... \cup A_{n}|= \sum_{i}|A_{i}|-\sum_{i<j}|A_{i}\cap A_{j}|+ \cdots + (-1)^{n-1}|A_{1}\cap A_{2}\cap\ \cap A_{n}|. $ ...
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Divisibility via Inclusion-Exclusion

Let $N$ be a large natural number, let $A$ be a subset of naturals, and ask: How many numbers $n\leq N$ are divisible by one or more numbers in $A$. This is a classical application of the ...
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How many ways can this quadrilateral be formed if no two of its vertices are next to each other?

A quadrilateral is formed by joining four vertices of a convex decagon. In how many ways can such a quadrilateral be formed if no two of its vertices are next to each other (that is, no two vertices ...
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Inclusion/Exclusion Probability for Devices

I'm having a hard time understanding the inclusion/exclusion principle. I was wondering if someone could help me out with this simple problem. Each of seven jobs in a computer system requests one ...
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156 views

Inclusion-exclusion principle: finding the number of solutions

Given the equation $\begin{cases} x_1+x_2+x_3+x_4 =18\\ 0\leq x_i\leq 7 \text{ with } x_i \in \mathbb{N} \text{ and } 1\leq i\leq 4 \end{cases}$ how do I calculate the number of solutions with the ...
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