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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How many 13-card hands have at least one Jack, King, Queen, or Ace?

So with this question, I came to this math: I have a J, Q, K, and A in four suits, and after having one of those face cards in a hand, now we are left to choose 12 more cards. so then I figure we get ...
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Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$

I am given this: Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$ if: $A \subseteq B$ and $B \subseteq C$ The sets are pairwise ...
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Probabilities of Unique Numbers in Roulette

To begin with I know there is a similar question asked and answered, but it is not what exactly i was trying to find (or at least it was only partially answered there) the question that i refer to is: ...
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37 views

Counting permutations with given condition

I need to find number of permutations $p$ of set $\lbrace 1,2,3, \ldots, n \rbrace$ such for all $i$ $p_{i+1} \neq p_i + 1$. I think that inclusion-exclusion principle would be useful. Let $A_k$ be ...
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How to make this inclusion-exclusion argument

I'm asked to count the number of functions $f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5\}$ such that $f(1)∉\{f(2),f(3),f(4)\}, f(2)\neq f(3), f(3) \neq f(4)$. How do I make the inclusion-exclusion argument ...
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Given $3$ types of coins, how many ways can one select $20$ coins so that no coin is selected more than $8$ times.

So I was given this question. Given $3$ types of coins, how many ways can one select $20$ coins so that no coin is selected more than $8$ times. First I make $x_1 + x_2 + x_3 = 20$ Then $ 0 \leq x_i ...
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Trouble with Inclusion-Exclusion (Multiplication Theorem)

$A_i$ is one event out of $n$. $$P\left(\bigcap_{i=1}^n A_i\right) = P(A_1)P(A_2|A_1) \dotsb P(A_n|A_1A_2...A_{n-1})$$ I have trouble with this theorem (I am not sure what its name is, so the title ...
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29 views

Where do the combinations come from in these examples of using the generalized inclusion exclusion principle?

I'm trying to understand where the combinations (the coeffecients of the $Si$'s) of this example come from. From my understanding, the first example denotes the generalized inclusion ...
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46 views

Why this is non negative

Assume you have a strictly decreasing non negative and convex function $f$ , and let $a_i$ for $i \in \{1,2,3\}$, be some positive real numbers, then $$g(t,a_1,a_2,a_3) = \sum_i^3 f(a_i t) ...
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Count the ways to choose distinct subsets $A_0, A_1, . . . , A_n$ of ${1, 2, . . . , n}$ such that $A_0 ⊂ A_1 ⊂ . . . ⊂ A_n$

I was given this question. Count the ways to choose distinct subsets $A_0, A_1, . . . , A_n$ of ${1, 2, . . . , n}$ such that $A_0 ⊂ A_1 ⊂ . . . ⊂ A_n$ I followed a different example to solve this ...
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3answers
40 views

Number of strings over a set $A$

How can I calculate the number of strings of length $10$ over the set $A=\{a,b,c,d,e\}$ that begin with either $a$ or $c$ and have at least one $b$ ? Is it accomplished through some sort of ...
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62 views

How many distinct, non-negative integer solutions are there for $2x_0+\sum_{i=1}^{m}{x_i}=n$?

We are given constants $m$ and $n$. How many non-negative integer solutions are there for $2x_0+\sum_{i=1}^{m}{x_i}=n$ satisfying the condition that$x_i\neq x_j$ if $i\neq j$? I thought a good first ...
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30 views

How many words are there of length $7$ are on $\{u,v,w,x,y,z\}$ without the string of letters $xxxx$?

How many words are there of length $7$ are on $\{u,v,w,x,y,z\}$ without the string of letters $xxxx$? My idea here is to use inclusion-exclusion. I was thinking of setting up the problem as ...
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Calculate Inclusion-Exclusion in Birthday Paradox

Follow-on from this post I was trying Birthday Paradox for 5-day calendar, with 3 people The probability that NONE of them have matching birthday is $5/5 * 4/5 * 3/5 = 0.48$ The probability there ...
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76 views

Total number of words possible using inclusion-exclusion

Calculate total number of words possible using alphabets A-Z having length 26 with following restrictions(Repetition not allowed) A and B should not occur adjacently (AB is not possible neither BA) ...
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340 views

Permutation of n objects with restriction of adjacent pairs

Given $n$ objects with values $\{x_1,x_2,x_3,\dots,x_n\},$ and $m$ pairs $a_k = \{x_i,x_j\}$. Let $\{p_1,p_2,p_3,\dots,p_n\}$ be a permutation of objects. The question is to find number of ...
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31 views

probability of coloring a $3\times 3$ table with two colors such that no $2\times 2$ square exists [closed]

imagine we have a $3\times3$ table ( or $3\times3$ square ) and we want to color each place of that $9$ places with two colors ( red and blue ). find the probability that no $2\times2$ square exists ...
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Three meetings where each attends exactly two

Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor of the group attended exactly two meetings. $21$ professors attended Mumbai meeting,$27$ ...
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32 views

Give a formula for $P(F\mid G^c)$ in terms of $P(F), P(G),$ and $P(FG)$ only

We know that $$P(F\mid G^c) = \frac{P(F \cap G^c)}{P(G^c)}$$ Thus, the denominator of our expression is just $1 - P(G)$ What is the numerator, however? I attempted to express the numerator using the ...
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Counting problem(principle of inclusion and exclusion)

Let C(n) denote the number of integers that are coprime to n from 1 to n, determine the value of C(36) and C(900) Please lend me a hand, I know principle of inclusion and exclusion but I can't figure ...
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Is there a formula for this case of infinite inclusion-exclusion?

Apparently, for any integers $m\ge 0,\ k\ge 1,$ and real $\alpha > 1$, the following series is convergent: $$S_{k,m}(\alpha) ...
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How many solutions does the equation $\sum_{i=1}^{k}{x_i}=c$ have, given that the $x_i\in\mathbb{Z}$ and $0\leq x_i\leq d$?

We are given initially some $k,c,d\in\mathbb{N}$. How many solutions $(x_1, x_2, ..., x_k)$ does the equation $\sum_{i=1}^{k}{x_i}=c$ have, where $x_i\in\mathbb{Z}$ and $0\leq x_i\leq d$?
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Number of 10-digit integers that have at least 3 different digits

I want to know how many 10-digit integers have at least 3 different digits, if leading 0 is not allowed. Ideally this question would help me learn the inclusion-exclusion principle. So I started ...
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44 views

How many integer solutions are there to $x_1 + x_2 + x_3 + x_4 + x_5 = 31$ with $x_i \leq i$

So i was given this question How many integer solutions are there to $x_1 + x_2 + x_3 + x_4 + x_5 = 31$ with $x_i \leq i$ So i looked at it and decided i have to use the inclusion exclusion ...
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How many permutations are there of $1, \ldots ,8$ in which none of the patterns 12, 34, 56, 78 appear?

I had thought this was a fairly simple Inclusion/Exclusion problem, but I noticed that my answer doesn't match the back of the book. Here is my thought process: Let S denote the set of all ...
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How many trees are there on 7 vertices, where vertices 2 and 3 have degree 3, 5 has degree 2, and all others have degree 1?

How many trees are there on 7 vertices, where vertices 2 and 3 have degree 3, 5 has degree 2, and all others have degree 1? So far, I am able to determine that vertices 1, 4, 6, and 7 are leaves. My ...
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Distribution of identical items vs distinct items

Given that there are $x$ ways to distribute $r$ distinct objects to $n$ distinct boxes (subject to any contraint, for example each box must have at least 1 object), and then we change the objects to ...
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proof of multi choose equivalence

I could really use some help proving this. Let n and k be positive integers, and let $\left(\!\!{n\choose k}\!\!\right) ={n+k -1\choose k}$, prove: $$\left(\!\!{n\choose k}\!\!\right) = ...
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Hat check problem. Ten friends total, five with sombreros, five with fedoras.

A group of ten people give their hats to the coatroom attendant. Five of the ten are wearing sombreros, and five and wearing fedoras. How many ways can the clerk return the hats so that no one gets ...
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28 views

The number of ways to chose $k$ children from a circle with $2n$ children

Let $k$ and $n$ be positive integers with $k \leq n$. Let $a(n,k)$ be the number of ways to place $k$ non-attacking rooks on an $n \times n$ board, where we exclude the tiles $\{(1, 1), (2, 2), ...
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Combinatorics submultisets Inclusion-Exclusion

Find the number of submultisets of {$25 \cdot a, 25 \cdot b, 25 \cdot c, 25 \cdot d$} of size $80$. I applied Inclusion-Exclusion to get; $$ {80+3\choose 3} - {4\choose1}\cdot{80-26+3\choose3} + ...
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Combinatorics Inclusion - Exclusion Principle

Find the number of integer solutions to $x_1 + x_2 + x_3 + x_4 = 25$ with $ 1 \leq x_1 \leq 6, 2 \leq x_2 \leq 8, 0 \leq x_3 \leq 8, 5 \leq x_4 \leq 9.$ Firstly, I defined $y_i = x_i - lower bound$ ...
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31 views

How many numbers are divisible by at least one element of $S$

I have a set S containing $N$ $(1 \le N \le 20)$ integers $a_i$ $(1 \le a_i \le 10^{18})$ and two numbers $a$ and $b$. I want to tell how many numbers between $a$ and $b$ are divisible by at least one ...
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Show that if n is a positive integer, then…where $d_n$ is the number of derangements of $n$ objects.

Show that if n is a positive integer, then $$n! = {{n}\choose{0}}d_n + {{n}\choose{1}}d_{n-1} + {{n}\choose{2}}d_{n-2}+\cdots+ {{n}\choose{n-1}}d_1 + {{n}\choose{n}}d_0 $$ where $d_n$ is the number ...
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Consider two sets $X$ and $Y$ with $|X| = m$ and $|Y| = n$ A function $f : X → Y$ is called almost-onto if $f$ misses at most one element of $Y$…

Consider two sets $X$ and $Y$ with $|X| = m$ and $|Y| = n$ A function $f : X → Y$ is called almost-onto if $f$ misses at most one element of $Y$. Find and prove a formula for the number of almost onto ...
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Use the Inclusion-Exclusion Principle to show that for any non-negative integers $m, r, n$ such that $m≤r≤n$…

Use the Inclusion-Exclusion Principle to show that for any non-negative integers $m, r, n$ such that $m≤r≤n$, $${n-m\choose{r-m}} = \sum_{i=0}^m(-1)^i {m\choose{i}} {n-i\choose{r}} $$ We have had ...
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How many permutations of $1,\ldots,8$ are there in which no even number appears in its natural position?

How many permutations of $1,\ldots,8$ are there in which no even number appears in its natural position? -Our teacher had us do a similar exercise to this one that looked for permutations without the ...
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33 views

Principle of inclusion and exclusion

There are 250 students who failed in an examination. 128 failed in maths, 87 in physics, 134 in aggregate. 31 failed in maths and physics, 54 in aggregate and maths, 30 in aggregate and physics. Find ...
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the number of objects from a collection of N objects that possess exactly m of the properties

Let $N_m$ denote the number of objects from a collection of $N$ objects that possess exactly m of the properties $a_1,a_2,\ldots,a_r$. Generalize the principle of inclusion-exclusion by computing ...
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There are 190 people on the beach. 110 are wearing sunglasses, 70 are wearing bathing suits, and 95 are wearing a hat.

There are 190 people on the beach. 110 are wearing sunglasses, 70 are wearing bathing suits, and 95 are wearing a hat. Everyone is wearing at least one of these items. 30 are wearing both bathing ...
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18 views

Number of permutations of [2n] where $x_i + x_{i+1} \ne 2n+1$

As stated in title, what is the number of permutations of $[2n]$ where $x_i + x_{i+1} \ne 2n+1 \;,\forall i\in[2n-1]$. I want to use the inclusion-exclusion theorem, and consider separately ...
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The inclusion-exclusion principle for probability and counting.

I am wondering whether "The inclusion-exclusion principle" from chapter 8 (rosen textbook) is the same with "The inclusion-exclusion principle for probability and counting" ?
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37 views

Arrangement of n*n chessboard with 3 colors and no single-colored rows or columns

How many ways are there to color a chessboard of dimension n*n with 3 colors so that there is no row and no column that has all of its grids colored the same color? Rotations count as separate ways.
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Couples sitting at a round table

Three couples are supposed to sit down at a round table such that no one sits next to his/her partner. The six seats are not numbered such that any rotation of a given configuration is regarded as ...
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Proving the principal of inclusion-exclusion using Möbius inversion

In Enumerative Combinatorics v. I, Stanley applies Möbius inversion to $B_n$, the poset of subsets of $[n]$, to show that $\forall f,g : B_n \to \mathbb{C}$, $$ g(S) = \sum_{T \subset S} f(T) \, \, ...
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Using the inclusion-exclusion principle to calculate the proportion of numbers divisible by factors

Let $m_1, ..., m_r$ be pairwise coprime numbers and $N=\prod\limits_{i=1}^r m_i$ . So I'm trying to calculate the proportion of the numbers 1 to N that are not divisible by any of the $m_i$, so I've ...
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1answer
66 views

How many permutations of letters are in “MATHISFUN” without words MATH , IS , FUN

I'm not sure I am doing this right: Using inclusion/exclusion principle S = MATHISFUN has 9 letters so this have 9! permutations. A = MATH , B = IS , C = FUN $\#A = 6!$ $\#B = 8!$ $\#C = 7!$ $ ...
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23 views

Deducing the number of divisors if we know their form

The task is to find out how many positive integers divide at least one of the given numbers: $10^{60}$, $20^{50}$, $30^{40}$. This is easily calculated using the inclusion-exclusion principle. ...
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23 views

Finding number of integers divisible by 2, 3 or 4 using inclusion-exclusion principle.

I want to find number of integers from 1 to 19 (both included) which are divisible by 2 or 3 or 4. Lets denote it by N. So counting and enumerating them gives N = 12. Integers are 2, 3, 4, 6, 8, 9, ...
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Inclusion-exclusion formula example

Compute the probability that a hand of 13 cards contains ace and king of at least one suit. I cannot understand why we use hear inclusion-exclusion formula. For the "al least" problems the common ...