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.
Questions tagged [inclusion-exclusion]
1500 questions
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Using Inclusion-Exclusion, probability of occurrence of exactly $m$ events out of $\{A_1,\ldots,A_n\}$
Using Inclusion-Exclusion, prove that the probability of occurrence of exactly $m$ events out of $\{A_1,\ldots,A_n\} = \sum_{i=0}^{n-m} (-1)^i \binom{m+i}{m} P_{m+i} $ where $P_k = \sum_{1 \leq i_1<\cdots \le i_k \leq n} P(A_{i_1} \cap\cdots \cap…
MathMan
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How many hexadecimal chains of numbers of length 6 are there with an "A" in the first position or with "F"...?
How many hexadecimal chains of numbers of length 6 are there with an "A" in the first position or with "F" in the 3rd and 4th position or a "0" in the last position? Only chains of numbers with at least one
hex value less than 9 should be counted.…
drey
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Inclusion–exclusion principle, find the number of students
There are ten students. Eight of them have travelled to Europe, seven of them speak Spanish and six of them study math. How many students have travelled to Europe, speak spanish and study math?
Well, I know that I have to find the intersection…
AaronTBM
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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 following way :
$\displaystyle\sum_{i=1}^N…
Mathejunior
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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
(similar to the one we have for $S-\{A_1\cup A_2…
2468
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Counting question based on Inclusion-Exclusion Principle
This was the question in one of the online test I was giving for preparation of an exam. I was not convinced by the final answer so I wanted to discuss it.
Consider a set S = {1000, 1001, 1002, ........ 9999}. The numbers in set ‘S’
that have…
user3767495
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Am I applying the Inclusion Exclusion principle correctly?
Q: Find how many integers $n$ satisfying $ 1\le n \le 5000$ are divisible by at least one of the numbers 4, 7 and 33.
I've done the following:
$$|A| = \lfloor 5000/4 \rfloor = 1,250 $$
$$|B| = \lfloor 5000/7 \rfloor = 714 $$
$$|C| = \lfloor 5000/33…
mal
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Number of Permutations of the Set $\{x_1, x_2, ... , x_n, y_1, y_2, ... y_n\}$ where $x_i, y_i$ are not next to each other
I'd like to find the number of permutations of the set
$$\{x_1, x_2, ... , x_n, y_1, y_2, ... y_n\}$$ where that $x_i, y_i$ are not consecutively located for each $i \in[n]$
solution
Let $A_i$ be the collection of cases where $x_i$ and $y_i$ are…
Beverlie
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Proof of $\sum_{r=0}^n(-1)^{n-r}\binom{n}{r}r^n = n!$
I want to provide proof for the following equation:
$$\sum_{r=0}^n(-1)^{n-r}\binom{n}{r}r^n = n!$$
I think it resembles the typical way of representing the Principle of Exclusion & Inclusion:
$$\sum_{\emptyset \ne J \in [n]}(-1)^{\mid…
Beverlie
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Inclusion-exclusion principle: Die thrown 10 times
Question:
Use inclusion-exclusion to calculate the probability that the numbers 1,2 and 3 each appear at least once when a die is thrown 10 times.
My attempt:
$(6^{10}-3.5^{10} +3.4^{10}-3^{10})/6^{10} $
Is this correct?
Mieke Möller
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Inclusion-Exclusion Principle; why is this wrong?
I'm unsure where I'm going wrong with this.
In a class of 40 people studying music: 2 play violin, piano and recorder, 7 play at least violina nd piano, 6 play at least piano and recorder, 5 play at least recorder and violin, 17 play at least…
OneGapLater
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Where am I wrong - Inclusion - Exclusion
Thanks for looking at the question.. I have tried to solve the problem but I'm missing something..
Problem:
How many solutions does the equation Xl + X2 + X3 = 13
have where X l , X2 , and X3 are nonnegative integers less
than 6?
I have…
Math Newbie
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How many ways can n couples stand in a line so that no one stands besides their spouse?
Trying to work out this problem:
The answer is the summation from $k = 0$ to $n$ for $(-1)^k \binom {n} {k} (2n - k)! \ 2^k $
Diante
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Sandwiches on Monday, Tuesday and Wednesday
15 students in a given class had a sanwich for lunch on Monday. 12 had a sandwich for lunch on Tuesday, and 9 had a sandwich for lunch on Wednesday. If 22 students had a sandwich at least once during these three days, what is the maximum number of…
space
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Inclusion-Exclusion Problem?
So I'm doing homework for my probability class and I am almost finished, however now I've got 1 Inclusion-Exclusion problem, which I have never done before.
A widget inspector inspects 12 widgets and finds that exactly 3 are defective. The widgets…
dj5
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