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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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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40 views

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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61 views

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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1answer
128 views

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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124 views

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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1answer
337 views

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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259 views

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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55 views

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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1answer
65 views

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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1answer
29 views

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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100 views

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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354 views

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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28 views

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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31 views

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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1answer
115 views

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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78 views

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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186 views

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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68 views

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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2answers
98 views

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 $\...
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1answer
74 views

Mixture of negative binomial distributions (technically some of them are geometric)

I have something that I am trying to compute. Let's say that a number is uniformly generated 1-4. What would be the expected number of generations required to get at least 3 1s and every other ...
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262 views

Inclusion/exclusion, at least and exactly arrangements?

The question is given the word "ARRANGEMENT", a) find exactly 2 pairs of consecutive letters? b) find at least 3 pairs of consecutive letters? I have the answer given from the tutor but it doesn't ...
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I don't see how the binomial theorem relates to the principle of inclusion and exclusion?

I'm learning discrete maths as a hobby at the moment and I got stuck when the tutor starting relating the binomial theorem to the principles of inclusion and exclusion. The video I was watching is ...
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481 views

Inclusion-Exclusion Principle for Three Sets

If $|A\cap B|= \varnothing $ (disjoint sets), then $|A \cup B|=|A|+|B|$ Using this result alone, prove $|A\cup B| = |A| + |B| - |A\cap B|$ $|A\cup B| = |A| + |B - A|$ $|A\cap B| + |B - A| = |B|$, ...
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Finding the maximum percentage of people who are not in A or B [closed]

60% of the population is in A 50% of the population is in B To get the maximum number of people in neither, is it right to find the maximum number of people in both (50% in this case) and plug that ...
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1answer
187 views

Number of ways to arrange $A,A,A,B,C,C$ such that no $2$ consecutive letters are the same

There is a question from my problem set that I am facing difficulty in solving. It says to find the number of ways to arrange $A, A, A, B, C, C$ so that no $2$ consecutive letters are the same. ...
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1answer
31 views

Answer check: What is the number of integers smaller than one million that contain two consecutive digits which are the same?

I try to use the Inclusion-Exclusion Principle to do this, with the component sets being $\{T_1, T_2, T_3, T_4, T_5 \}$, where $T_i$ denotes the set of 6-digits integers containing repeated digits at ...
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1answer
84 views

Number of ways to roll $S$ with $n$ dice.

I have been working on this problem for the past 3 days and I am not having a lot of luck with it. I posted part of a) here already and I got some very useful advice. I am sitting a test in this ...
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2answers
76 views

I need to find the combinatory formula for this set.

Problem 1. Fix a positive integer $n$. For every integer $S \geq n$, let $N_{n,S}$ denote the number of possible ways in which a sum of $S$ can be obtained when $n$ dice are rolled. For example, for $...
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2answers
72 views

Validation of Answer via Truth Table as Reason in Isolation

I've tried various google and math.SE search strings but I'm having trouble formulating a query that gives me relevant information. Questions Does this table below accurately represent an acceptable ...
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1answer
42 views

Principle of Inclusion and Exclusion Notation [closed]

Let $A,B,$ and $C$ be sets such that $A\subset C, B\subset C$, with $|C|=n$, $|A|=x$, $|B|=y$, and $|A\cap B|=z$. What is $|C\setminus(A\cup B)|$?
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79 views

How many $10$-digit numbers (allowing initial digit to be zero) in which only $5$ of the $10$ possible digits are represented?

The answer I found was $$(5^{10}-|\text{only}~4~\text{digits}|-|\text{only}~ 3|-|\text{only}~ 2|-|\text{only}~1|) \cdot C(10,5)=$$ where $|\text{only}~1~\text{digit}| = 1^{10} \cdot C(10,1)$ $|\...