# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...