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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inclusion/exclusion for permutations from 1 to 7 with conditions

"Question: Find the number of permutations of 1, 2, 3, 4, 5, 6, 7 that do not have 1 in the first place, nor 4 in the fourth place, nor 7 in the seventh place. Ans:We use the inclusion-exclusion ...
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Non negative integral solutions to a linear equation with constraints [closed]

Consider the equation $p+q+r+s=49$, find all the non-negative integral solutions with constraints $0\le p \le 5$ and $0 \le q \le 10$. My book says to use inclusion exclusion, is there any other way ...
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Why does the method of inclusion/exclusion give the wrong answer when finding the number of integers b/w 1 and 10 that are not divisible by 2,3 or 5?

Let $S=\{1,2,\dots, 10\}$. METHOD 1: I'm first counting the integers that are divisible by $2, 3$ or $5$ in $S$ and then subtracting from the total as follows: Let $A, B, C$ be the set of integers ...
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When should Sylow subgroups intersect and when they should not?

Here is the question I am trying to understand its solution: Prove that a group of order $11 \times 2^{10}$ has a normal subgroup. And here is a solution I found to the part of excluding the case $n_2 ...
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There are $n$ points inside a $1\times1$ square that can form a convex polygon. Can $3$ of them exist, such that $\mathcal{A}\le\frac{8}{n^2}$?

There are $n$ points inside a $1\times1$ square that can form a convex polygon. Can $3$ points of them exist, such that $\mathcal{A}\le\frac{8}{n^2}$? where $\mathcal{A}$ area of triangle formed by ...
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2 answers
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Do the terms of an inclusion-exclusion summation decrease?

If $|A_1 \cup A_2 \cup \ldots \cup A_n| = c_1 - c_2 + \ldots + (-1)^n c_n$, where $c_i$ is the sum of the sizes of all of the intersections of $i$ sets at a time (inclusion-exclusion principle); i.e $...
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$n$ people place their phone into a bag. Each person randomly picks a mobile phone from the bag [duplicate]

The question is split into two parts. Determine an expression for the probability that at least one individual selects their own mobile phone. Let $L(n)$ denote this expression. Determine the exact ...
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2 votes
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An inequality involving the Inclusion-exclusion principle

Given a probability space $(Ω, \mathfrak{A}, P)$ and events $A_1,A_2 . . . , A_n ∈ \mathfrak{A}.$ Show that $$P\bigg(\bigcup_{j=1}^{n} A\bigg) \leq \min_{1\leq i\leq n} \bigg( \sum_{j=1}^{n}P(A_j)- \...
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Combinatorics explanation of Inclusion-Exclusion Principle Exactly-$m$ Properties Formula $E_m=\sum_{j=m}^n(-1)^{j-m}\color{blue}{{j\choose m}}S_j$

I'm trying to "explain" (I think this would not be a formal proof because I use a special case of the formula itself when I was "proving" it. So the tag I put might need to be ...
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Is there any information that is missing in this problem?

Q) An advertising agency finds that, of its 200 clients who use Television or Radio or both, 150 use Television. How many use only Radio? A). 150, B). 100, C). 50, D). Data is insufficient According ...
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Probability of rolling at least one 5 before the first 6

The question has two parts and is: Suppose you repeatedly roll a fair six-sided die with numbered faces 1 to 6. Determine the following The probablity the $n^\text{th}$ roll produces the first 6 You ...
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Inclusion-exclusion algorithm for sets of multiples

Given a finite set $S$ of positive integers, I'd like to compute the natural density of their multiples. (This density always exists, even for infinite sets.) This can be found with straightforward ...
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Montmort’s matching problem with venn diagram

is it possible to have a venn diagram represent de Montmort matching problem where the number of cards is $n=3$ with the elements included in the diagram? I understand how inclusion/exclusion works(...
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Inclusion–exclusion principle; what is $(-1)^{n+1}$

could somebody kindly confirm that my understanding of inclusion-exclusion matches it's formula. for a 3 sets example; we add 3 unions, subtract the total of all 3 pairwise intersections and add the ...
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1 answer
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Choosing $n$ out of $k$ classes limited to $r$ for each class

As in the title, the problem is to choose $n$ elements from $k$ classes, in a way that from each class there are at most $r$ elements chosen (that is, from $0$ to $r$, inclusive). The specific ...
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Why is $\sum_{i=0}^k (-1)^{k-i} {n \choose i} {n-i-1 \choose k-i}=1$

Prove $\sum_{i=0}^k (-1)^{k-i} {n \choose i} {n-i-1 \choose k-i}=1$. I was recently working through the proof of a question using inclusion exclusion and was stuck on this part. I verified this ...
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4 votes
1 answer
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binomial coefficient in the inclusion-exclusion principle

Suppose n people leave their coats at the cloakroom, but on leaving the cloakroom the supervisor randomly gives any coat back to each person. Now to determine the number of permutations in which s ...
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2 votes
1 answer
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Number of ways to color a square graph with diagonal line using principle of Inclusion and Exclusion

How many ways can we color the graph if adjacent vertices receive different colors ? It's easy to see that the answer is $n(n-1)(n-2)^2 = n^4-5n^3+8n^2-4n$, where $n$ is the number of colors (fix a ...
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Determine the number of graphs.

For vertex set $\{1, 2, \dots, n\}$, how can I find the number of graphs having no component with exactly two vertices? I thought about the number of graphs in an ignition method, and I was worried ...
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7 votes
5 answers
201 views

Find the number of rearrangements of AABBBCCDDD where there are no two consecutive As or Bs

I tried using inclusion exclusion where set $A$ is all the rearrangements where two $A$s are consecutive and set $B$ where two $B$s are consecutive. However, I got $8400$ which is incorrect. I think ...
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How many $5$ letter words can be made from $15$ letter set where multiple conditions must be met

a) How many $5$-letter words can be made using letters from the $15$ letter set $\{A, B, C ... , O\}$ such that the letters are all different and in alphabetical order? b) How many are there if we add ...
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6 votes
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Combinatorial interpretation of a sum

I would like to know if there exists a way to interpret this sum by a combinatorial argument $$\sum _ { k = 0 } ^ { n } \frac { ( - 4 ) ^ { k } k ! } { ( 2 k + 1 ) ! ( n - k ) ! } = \frac { 1 } { ( 2 ...
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How many solutions does $x_1+x_2+x_3 = 11$ have if $0\le x_1 \le 3$, $0\le x_2 \le 4$, and $0\le x_3 \le 6$?

How many solutions does $x_1+x_2+x_3 = 11$ have if $0\le x_1 \le 3$, $0\le x_2 \le 4$, and $0\le x_3 \le 6$? I tried to do it with method 2 but there is a problem; \begin{align*} x_1+x_2+x_3 = 11\tag{...
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1 answer
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Is $X_{0}$ a subset of $X_{0}\sqcup X_{1}$?

Since the disjoint union of $X_{0}= \{x_{1}, x_{2}, x_{3}\}$ and $X_{1}= \{x_{1}, x_{2}\}$ is: $ X_{0}\sqcup X_{1}= \{(x_{1},0), (x_{2},0), (x_{3},0), (x_{1},1), (x_{2},1)\}$ I have to ask; is $X_{0}$ ...
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Question about Inclusion-Exclusion Principle

I have a set of independent events $\{B_1,B_2,...,B_k\}$ where $k\leq n$ for some integer $n>0$. I know the following probabilities $$ p(B_i) = (1-r^{\frac{n}{n-i}})\\ p(\overline{B_i} \cap B_j ) =...
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how to find the number of integers between two numbers inclusive that are divisible by any of the two numbers X or Y. [closed]

You are given 4 integers X, Y, L, R. You need to find the number of integers between L and R inclusive that is divisible by any of the two numbers X or Y. How to find the answer without trying all ...
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prove if $E_{1},E_{2},...,E_{n}$ are independent events then $p(\cup_{i=1}^{n}E_{i})=1-\prod_{i=1}^{n}[1-p(E_{i})]$

prove if $E_{1},E_{2},...,E_{n}$ are independent events then: $p(\cup_{i=1}^{n}E_{i})=1-\prod_{i=1}^{n}[1-p(E_{i})]$ my try: $$ p(\cup_{i=1}^{n}E_{i})=1-p((\cup_{i=1}^{n}E_{i})^{c})=1-p(\cap_{i=1}^{n}...
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What is the probability of choosing a 4-digit number that starts with 1 or 2, and has at least 3 of the same digit?

This is a question that I haven’t been able to solve! Some more information: The four digit number must start with either 1 or 2. I’ve assumed that combinations is required, as the order of the final ...
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Stars and Bars vs. PIE for probability question.

The given question was: An airport bus drops off 35 passengers at 7 stops, each passenger is equally likely to get off to any stop and the passengers act independently of one another, the bus makes a ...
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1 vote
2 answers
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How many ordered pairs $(A,B)$ there are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5,\ldots,20 \}$ such that $A \cap B = \{2,5,6\}$?

How many ordered pairs $(A,B)$ there are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5,\ldots,20 \}$ such that $A \cap B = \{2,5,6\}$? Solution : Any number from $\{ 1,3,4,7,\ldots,20 \}$ can go either ...
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How many strings of five decimal digits must be starting or ending with an odd number?

How many strings of five decimal digits must be starting or ending with an odd number? Everywhere, I looked over the internet they used this method: Thus, number of ways $=5 \cdot 10 \cdot 10 \cdot ...
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1 vote
1 answer
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If 4 cards are picked from the 36 number cards in a deck, how many ways can you choose the cards so that no two are consecutive numbers?

The number cards range from 2-10; there is 4 of each card, and 36 in total. The order of the cards matters: the order in which they are chosen is the order in which they are displayed. No consecutive ...
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Calculating a 3 Circle Venn Diagram only knowing A, B, and C?

I came across this question: If there are 40 students in a class, 30 of them got A in Music, 33 of them got A in PE, and 37 of them got A in Art, at least how many students got all 3 As? The first ...
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Weighted Version of Inclusion-Exclusion Principle.

I am trying to understand the formula $\omega(E_{s}$)=$\sum\limits_{t=s}^{r}(-1)^{t-s}\binom{t}{s}W(t)$, where E$_{s}$={k|f(k)=s} and f is f=$\chi_{N_{2}}+\cdot\cdot+\chi_{N_{r}}$, where $\chi$ is the ...
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In how many ways we can place 9 different balls in 3 different boxes such that in every box at least 2 balls are placed?

In how many ways we can place $9$ different balls in $3$ different boxes such that in every box at least $2$ balls are placed? Approach 1: Let $x_1$, $x_2$, $x_3$ denote the number of balls in boxes $...
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4 votes
1 answer
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Formulating an alternating sum of product combinations

Consider some list $A=(a_1,a_2,\cdots,a_n)$. I'd like to find a closed form for the following operation. $$f(A)=\sum_{k=1}^n(-1)^{k-1}s_k= s_1-s_2+\cdots(-1)^{n-1}s_n.$$ Where $s_k$ is the sum of all ...
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1 vote
1 answer
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How many 12-letter words are there with no block $5 \times a$, $4 \times b$ and $3 \times c$

We arrange 12-letter words having at our disposal five letters $a$, four letters $b$ and three letters $c$. How many words are there without any block $5 \times a$, $4 \times b$ and $3 \times c$. I ...
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2 votes
2 answers
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To find the number of ways to put $14$ identical balls into $4$ bins with the condition that no bin can hold more than $7$ balls.

To find the number of ways to put $14$ identical balls into $4$ bins with the condition that no bin can hold more than $7$ balls. I have tried the following: The total no of ways to distribute $14$ ...
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1 vote
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Why inclusion-exclusion principle fails for vector subspaces

Let $U,V,W$ be three vector subspaces of a same vector space. It is well-known that $$\dim (U + V) = \dim U + \dim V - \dim (U \cap V)$$ works but $$ \dim(U +V + W) = \dim U + \dim V + \dim W - \dim (...
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Finding number of nonnegative solutions for the equation $x_1+ x_2+ x_3+ x_4+ x_5=9$ when $x_1 \geq 1, x_5 \leq 5$

I need to find the number of nonnegative solutions for the equation $$x_1+ x_2+ x_3+ x_4+ x_5=9$$ when $x_1 \geq 1, x_5 \leq 5$. I know I need to use $\binom{n+k-1}{k-1}$ and Inclusion–exclusion, but ...
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Inclusion–exclusion principle problem - distributing sweets

What is the number of all possible ways of distributing 16 DIFFERENT candies in a group consisting of 2 male and 4 female students if each male student receives at most 3 sweets? I have no idea how to ...
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Probability of the first card of a standard deck being a king or a heart.

I am solving an equation to find the probability that the first card of a standard, 52 card deck is a king or a heart, and I feel like I’m doing this wrong. I set up this equation (k for king, h for ...
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1 vote
1 answer
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Permutations: play a song 3 times before all songs have played at least once.

A playlist has $10$ songs. Two musics players ($A$ and $B$) implement the "shuffle and repeat" feature a little differently. In $B$, no song will be played twice in a row, but it is possible ...
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1 vote
0 answers
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Distribute $r$ indistinguishable balls randomly into $n$ cells. The probability that exactly $m$ cells will contain exactly $k$ balls each is? [duplicate]

Distribute $r$ indistinguishable balls randomly into $n$ cells. The probability that exactly $m$ cells will contain exactly $k$ balls each is? The answer: $$\frac { m!r!} { m!n^{r} } \begin{equation}...
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4 votes
6 answers
127 views

Revisit : $20\choose 5$ subsets without 3,4 or 5 consecutive numbers

Addendum-2 just added to my question. Addendum just added to my question. $\underline{\textbf{Overview}}$ This is a self-answer question of this original question. I strongly suspect that the ...
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-3 votes
2 answers
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How many solutions are there to $x_1+x_2+x_3+x_4=10$ such that for all $0\le x_i\le3$? [closed]

I know the general path is that k=10 and n=4, so it somehow relates to ${13 \choose 3}$. I know I'm missing a part due to the constraint that all elements are between 0 and 3. For now, $${13 \choose 3}...
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2 answers
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Ways to divide $14$ numbered balls into $3$ non-empty groups

Fourteen numbered balls (i.e., $1$, $2$, $3$, $\ldots$, $14$) are divided in three groups randomly. Find the probability that sum of the numbers on the balls, in each group, is odd. There is this ...
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0 votes
1 answer
52 views

Number of colorings using inclusion–exclusion principle

Let's say there are $21$ possibilities to color an object using an inventory of the colors red, green and blue. (By 'inventory' I mean that we don't have to use all three colors, we can also only use ...
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2 votes
0 answers
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Alternative way to deduce combinatorial identity?

By considering the partial fraction decomposition of a certain family of expressions, I have managed to deduce the identity: $$\sum_{k=0}^{n} (-1)^k {2n\choose k} (n-k)^{2m} \equiv 0 $$ where $n > ...
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2 votes
1 answer
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Principle of Inclusion-Exclusion with Substitution Ciphers

Consider an alphabet with $2n$ symbols and the substitution cipher that maps $p_i$ to $c_i$ for all $i$. If the numerical representation of $p_i = i$ for every $i$, how many substitution ciphers exist ...
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