# 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 ...
1 vote
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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 ...
1 vote
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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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### 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{...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
1 vote
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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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### 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 $14$ ...
1 vote
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### 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 ...