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Questions tagged [inclusion-exclusion]

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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Counting with Principle of inclusion-exclusion

Let $n$ be a positive integer. Find the number of permutations of $ (1, 2,...,n)$ such that no number remains in its original place. Solution: To do this, first we are going to count the number of ...
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Combinatorics Inclusion–exclusion principle [on hold]

3 man goes into a restaurant with a hat and umbrella each man takes 1 hat and 1 umbrella on the way out. 1. How many options are there for all of them to leave without their hat and umbrella? 2.How ...
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What is the probability that no cup is on a saucer of the same colour?

A tea set comprises four cups and saucers in four distinct colours. If the cups are placed at random on the saucers, what is the probability that no cup is on a saucer of the same colour? MY ATTEMPT ...
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In how many ways can letters in a word CALCULUS be rearranged

In how many ways can letters in a word CALCULUS be rearranged such that no 2 same letters stand next to each other. I’ve been thinking of Inclusion-Exclusion principle. Is there any different way to ...
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Number of ways of coloring n objects which are laid in a row with k colors such that the adjacent objects are of different colors

Given n objects, which are lying in a straight line next to each other, in how many ways we can color them with k colors (all must be painted) such that the adjacent boxes not of same colors. I can ...
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Given $a_1, a_2, \cdots, a_N$ such that $a_1+a_2+\cdots +a_N = S$ for some given $S$, find the number of ways such that someone is $\geq T$.

Given $a_1, a_2, \cdots, a_N$ such that $a_1+a_2+\cdots +a_N = S$ for some given $S$, find the number of ways such that someone is $\geq T$. The question is solved using Inclusion Exclusion in the ...
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A Question on the Inclusion-Exclusion Principle in Probability Theory

Let $n \geq 3$ be a fixed integer and consider an arbitrary set of the events $\{A_i, 1 \leq i \leq n \}$ in the same probability space. Consider the following two sums of probabilities: \begin{align} ...
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pigeonhole principle, Number of consecutive visiting days… [closed]

A social worker has to make altogether 60 visits, at least one on each day. Is there a period of consecutive days on which he makes exactly 15 visits if he makes his visits on a) 31 days b)30 days? I ...
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Derangements $p$ of $1,2,\dots,n,n+1$ such that $n+1$ doesn't go to $n$

Recall that the number or Derangements of $1,2,\dots,n$ is a permutation $p$ such that $p(i) \neq i$ for all $i$. We can express it with the recurrence $D_n=(n-1)(D_{n-1}+D_{n-2})$ or by the closed ...
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Inclusion exclusion principle counting

50 students traveled to Europe, last year. Of these, 12 visited Amsterdam, 13 went to Berlin, and 15 were in Copenhagen. Some visited two cities: 3 visited both Amsterdam and Berlin, 6 visited ...
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Number of one-one function from sets $A$ to $B$.

Let $A=\{1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5\}$. Number of one-one function from $A$ to $B$ such that $f(1) \neq 0$ and $f(i)\neq i$ for $i={1,2,3,4,5}$ is _______ . So I know one one function means ...
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Prove that $(1-x)^n \leq 1 -xn+\frac{n(n-1)}{2}x^2$ when $0\leq x\leq 1$ and $n\geq2$

Reading a book I saw this inequality $$ (1-x)^n \leq 1 -xn+\frac{n(n-1)}{2}x^2 $$ when $0\leq x\leq 1$ and following the author it descended from the inclusion-exclusion principle. I don't understand ...
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Seating Arrangement with Derangement

A group of n students is assigned seats for each of two classes in the same classroom.How many ways can these seats be assigned if no student is assigned the same seat for both classes? Okay so this ...
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How many ways are there to divide n dancers into dance circles where in each circle num of dancers >=2?

Question:Be n\geq 2, how many ways are they to divide n dancers to circles so each circle has at least 2 dancers? I saw a similar question here but it was where order matter in the circles and ...
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Letters and Envelops problem

Consider a machine whose job is to place 100 letters into 100 envelops.The machine is defective and makes mistakes.What is the probability that in a group of 100 letters no letter is put into the ...
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Explanation of counting by Inclusion Exclusion

In my notes I have the following as an example for counting by inclusion exclusion. Let S be a set. Let $c_i(x)$ where $i=1,2,3,4....k$, be a statement that is either true or false for $x \in S$. ...
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Let 𝐴, 𝐵 and 𝐶 be sets. Prove formally that |𝐴 ∪ 𝐵 ∪ 𝐶| = |𝐴| + |𝐵| + |𝐶| − |𝐴 ∩ 𝐵| − |𝐴 ∩ 𝐶| − |𝐵 ∩ 𝐶| + |𝐴 ∩ 𝐵 ∩ 𝐶|

By using a Venn diagram we can see almost immediately that the cardinality of the members of the equality is in fact the same, however the exercise asks me to prove it formally and there is where my ...
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Inclusion Exclusion Principle with mixed lower and upper bounds

I know how to solve most types of these problems, but this one is a bit different. Problem: John is getting his friend some balloons for his birthday. He can have 4 types of colors (red, blue, ...
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Determine the number of integer solutions to $x_1 + x_2 + x_3 + x_4 = 19$, where $−5 \le x_i \le 10$ for all $1 \le i \le 4$

What I have so far: Goal: Using the inclusion exclusion I want to find $|\overline A_{1}\cap \overline A_{2} \cap \overline A_{3} \cap \overline A_{4}| = |U| - S_{1} + S_{2} - S_{3} + S_{4}$ $S_{k} ...
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$n$ guests, each guest brings a prize, how many ways may the prizes be given out so nobody gets the prize that they brought?

Each person attending a party has been asked to bring a prize. The person planning the party has arranged to give out exactly as many prizes as there are guests, but any person may win any number of ...
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Combinations of flowers using the counting method for integer partitions.

I have this problem to complete that wants to know how many combinations of flowers can there be in a bouquet of 25 flowers, such that: $r+c+d+t=25$ where $r=$roses, $c=$carnations, $d=$daisies and $...
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What is the number of arrangements in the word “EDUCATION” where vowels are never together?

I know the answer is $$ 4! P(5,5) $$ Because we can arrange the consonants amongst themselves in 4! ways and then independently insert the five vowels into the five spaces available. My question ...
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Inclusion exclusion and partition of a set - making sure I understand the concepts

If I may, I would like to verify my solution of a couple of homework questions, and by doing so asking a few questions about these topics. Let $X$ be a set of size $n$. How many distinct triplets $(A,...
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Counting ways to line up for a family photo.

Question: A family lines up for a photograph. In each of the following situations, how many ways are there for the family to line up so that the mother is next to at least one of her daughters? The ...
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For positive integers between 999 and 100 inclusive, how many contain the digit 5?

The question comes in two parts: For positive integers between 999 and 100 inclusive, how many contain the digit 5 at least once? For positive integers between 999 and 100 inclusive, how many contain ...
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Person with seven friends invites subset of three for one week

Suppose that a person with seven friends invites a different subset of three friends to dinner every night for one week (seven days). How many ways can this be done so that all friends are included at ...
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Ways to seat $n$ couples around a circular table with restriction [duplicate]

How many ways are there to seat $n$ couples around a circular table such that no couple sits next to each other? I know that since there are $n$ couples, there are $2n$ people. I need to use ...
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Combinatorics Question Involving Arranging Letters in INTELLIGENT with Restriction [duplicate]

How many ways are there to arrange the letters in INTELLIGENT with at least two consecutive pairs of identical letters? I know that we would use the inclusion exclusion principle here and that there ...
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Combinatorics Question Asking How Many Integer Solutions Given Different Restrictions

How many integer solutions of $x_1 + x_2 + x_3 + x_4 = 28$ are there with: (a) $0 ≤ x_i ≤ 12$? (b) $−10 ≤ x_i ≤ 20$? (c) $0 ≤ x_i, x_1 ≤ 6, x_2 ≤ 10, x_3 ≤ 15, x_4 ≤ 21$? \ \ My attempts (I was ...
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Doubt regarding two permutations and combinations problems

Here are two questions from permutations and combinations: There are 10 identical blankets. These are to be distributed among 4 (distinct) beggars. In how many ways can you do this so that each ...
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How many integer solutions of $x_1$ + $x_2$ + $x_3$ + $x_4$ = 28 are there with [duplicate]

How many integer solutions of $x_1$ + $x_2$ + $x_3$ + $x_4$ = 28 are there with (a) 0 ≤ $x_i$ ≤ 12? (b) −10 ≤ $x_i$ ≤ 20? (c) 0 ≤ $x_i$, $x_1$ ≤ 6, $x_2$ ≤ 10, $x_3$ ≤ 15, $x_4$ ≤ 21? I have tried....
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How to show $ \mathbb P(B \cap C) * \mathbb P(B\cup C)$?

Let B,C be events in a probability space. Show $ \mathbb P(B\cup C) \mathbb P(B\cap C) \leq \mathbb P(B) \mathbb P(C) $. My work: I started to use inclusion- exclusin principle on the left side, but ...
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Value for $\Bbb P (A \cup B \cup C) $

Let $A,B,C$ be events in a probability space. Suppose that $\mathbb P(A) = 1/4\\ \mathbb P(B^c) = 2/3\\ \mathbb P(C) = 1/2\\ \mathbb P(A^c\cap B) = 1/4\\ \mathbb P(A\cap C) = 0\\ \mathbb P(B^c\cup C^...
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How many compositions does the integer $12$ have into three parts none of which is equal to $2$?

I want to find the number of compositions that satisfy the the following conditions: $x_1 +x_2 +x_3 = 12$ and $x_i \neq 2$ Total $\binom{14}{2}$ compositions (weak) Number of compositions where one ...
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Proving real inclusion for $1 \leq p \lt r \lt \infty$

How can I show for $1 \leq p \lt r \lt \infty$ that there is the real inclusion {${x \in \mathbb{R^n}: \left\lVert x \right\rVert_p \leq 1} $} $\subsetneq$ {$x \in \mathbb{R^n}: \left\lVert x \...
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Inclusion_exclusion general formula for intersections?

Assume $A_1,\, A_2, \ldots , A_n$ are subsets of a finite set $S$. Can we find an expression for the size of $S-\{A_1\cap A_2 \cap \ldots \cap A_n\}$ in term of the unions of any number of $A_i$'s (...
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Combinatorics question involving distributing 9 different candies to three different kids

Nine different chocolate bars are to be distributed to 3 different kids. a) In how many ways can this be done if there are no restrictions? b) In how many ways can this be done if the child A ...
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On the probability of placing cups on saucers

A tea set consists of six cups and saucers with two cups and saucers is each of the three different colours. The cups are placed randomly on the saucers. What is the probability that no cups is on a ...
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Principle of Inclusion and Exclusion: Smallest Possible Number of Students in the Room [closed]

In a classroom, 9 students are talking, 5 are standing, and 4 are reading. 1 student is standing and not talking. 1 student is reading and not talking. What is the smallest possible number of ...
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Arranging 3 types of balls

Say we have $3n$ balls of 3 types: 1,...,n big balls 1,...,n medium sized balls 1,...,n small balls I'd like to arrange them in triples so that every triple contains one of each type, but in each ...
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Probability a student speaks a language given

Question Every student in a class speaks at least one of three languages. For every language, the probability that a random student speaks that particular language is $\frac{3}{4}$ and for every ...
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Number of nine-digit strings that contain each of the odd digits

Find the number of nine-digit strings that contain each of the odd digits $1, 3, 5, 7, 9$. We can use any digit from $0,1,...,9$ but it is must include all odd digits. Repetitions are allowed. I ...
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Combinatorial meaning of $L_m=S_m-{m\choose m-1}S_{m+1}+{m+1\choose m-1}S_{m+2}\mp\dots+(-1)^{n-m}{n-1\choose m-1}S_n$

I want to understand the meaning behind the coefficients of the following formula, $L_m$: Let $U$ be a finite set, and there are $n$ properties defined on it: $a_1,a_2,\dots,a_n,$ and let $S_m$ ...
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Calculate probability using inclusion-exclusion and deduce formula for binomial coefficicient

We choose uniformly a group of $k$ people selected from $n$. For $m \leq k$, calculate using inclusion-exclusion the probability that $m$ special people are in the group and then deduce that \begin{...
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Proof of Poincare's Inclusion-Exclusion Indicator Function Formula by Induction

The Poincare's inclusion exclusion formula is given by \begin{align} \Bbb{I}_{\bigcup_{1\leq j\leq n}A_j}=\sum_{1\leq j\leq n}\Bbb{I}_{A_j}+\sum^{n}_{r=2}(-1)^{r+1}\sum_{1\leq i_1<i_2<\cdots<...
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Demonstration inclusion - exclusion by induction for n elements.

I am trying to demonstrate the principle of inclusion - exclusion for n elements. For n = 2 and n = 3 I have calculated it applying properties of monotonicity, dimension, complementarity and it gives ...
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Inclusion-Exclusion principle / Coupon Collector

Here's the setup to the problem. Assume there are $5$ types of coupons, and you keep collecting coupons until you meet your condition. What is the probability that the number of trials needed to ...
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Three people take a series of exams, three grades given for each exam. Who placed second in Geometry?

Alice, Betty, and Carol took the same series of examinations. For each examination there was one mark (grade) of $x$, one mark of $y$, and one mark of $z$; where $x, y, z$ are distinct positive ...
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Find the number of permutations of the 8 letters AABBCCDD, taken all at a time, such that no two adjacent letters are alike.

This appears to be an inclusion/exclusion problem. My first step was to find the total permutations with no restrictions, using $\frac{8!}{2!2!2!2!} = 2520$. What would be the permutation formulas ...
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What does the notation $ a \to a : A \to B$ means in the context of the word “inclusion”?

I read the following sentence (from these notes on logic): If $\mathcal A \subseteq \mathcal B$, then the inclusion $ a \to a : A \to B$ is an embedding $\mathcal A \to \mathcal B$ where $\...