# 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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### Inequality of inclusion-exclusion terms

While analyzing the properties of an algorithm I am working on, I came up with the following inequality of inclusion-exclusion terms. Let $0 \leq i \leq j < k$ be natural numbers. Then, I want to ...
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### In how many ways can 5 non identical letters be mailed if there are 3 different mailboxes available if each letter and no mailbox remains empty [closed]

In how many ways can 5 non identical letters be mailed if there are 3 different mailboxes available if each letter and no mailbox remains empty?
1 vote
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### What is the exact probability of rolling every possible result on an a-sided die at least b times each after rolling c times?

I have found answers to similar, much simpler questions on this site before (such as here How do you calculate probability of rolling all faces of a die after n number of rolls?), but can't find ...
1 vote
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### Practical illustration of the inclusion-exclusion principle

I am currently doing a combinations problem that has two parts. Part A: There are $26$ uppercase letters in the alphabet (A, B, C ... Z). How many $20$-letter words contain the subword 'DOWNTIME' ...
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### In how many ways $n$ balls can be given to $k$ children so that no child gets more than $3$ balls when balls are distinct/identical?

In how many ways $n$ balls can be given to $k$ children so that no child gets more than $3$ balls when balls are a) balls are distinct b) balls are identical I am trying to solve this problem using ...
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1 vote
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### How many permutations of set are there with 4 fixed points?

Given the set of numbers $A = \left \{ 1, 2, 3, 4, 5, 6, 7, 8 \right \}$. How many permutations leave exactly four numbers fixed? I've considered this as a solution... but then I stumbled upon a ...
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### If 5 married couples are arranged in a row, find the probability that no one sits next to their spouse? [duplicate]

I tried to first finding the probability that only one couple sits together (9x2!8!) then two couples (25x2!2!2!6!) then three (22x3!2!2!2!4!) then four (14x4!2!2!2!2!) then five (5!x2!2!2!2!2!) I ...
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### proof for $\sum_{r=0}^{40} {40 \choose r} (-1)^r {100-r \choose 40} = 1$

$\sum_{r=0}^{40} {40 \choose r} (-1)^r {100-r \choose 40} = 1$ I have tried expanding, simplifying it. It doesn't work. Anyone has any ideas?
1 vote
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### We roll a $6$-sided die $n$ times. What is the probability that all faces have appeared?

How do I solve these questions, there are so many cases I know the total outcomes is$6^n$. However, I am lost after that.
1 vote
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### Number of students who study both Hindi and English

In a survey of $100$ students, the number of students studying the various languages is found as: English only $18;$ English but not Hindi $23;$ English and Chinese $8;$ Chinese and Hindi $8;$ ...