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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

How many integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 32$ with $0 \leq x_i \leq 10$ for $i = 1, 2, 3, 4$? How many integer solutions are there to the inequality $$ y_1 + y_2 + y_3 + y_4 < 184 $$ with $y_1 > 0$, $0 < y_2 \leq…
infinity
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Distribute $10$ distinct prizes to $4$ students

Inclusion Exclusion Used Theorem 8.1 The Question: In how many ways can one distribute $10$ distinct prizes to $4$ students with exactly $2$ students getting nothing. b) How many ways have at least 2 students receiving prizes? My Work: Let $S =$…
Dunka
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Calculate the cardinality of the union of all two-intersections

Suppose we have the classic inclusion-exclusion problem setup: we have a superset $S$, and $k$ sets $A_1,A_2,\dots,A_k$. We know $|S|, \forall_i (|A_i|=c_1), \forall_{i,j}(|A_i\cap A_j|=c_2), \dots, |\bigcap_{i=1}^k A_{i}|=c_k$. Now I want to…
Nick
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Show that $\sum\limits_{P\subseteq S}(-1)^{n-|P|}\left(\sum\limits_{a\in P}a\right)^n=n!\prod\limits_{a\in S}a$, where $n=|S|$

I want to prove the following: $$\sum_{P\in 2^S}(-1)^{n-\mid P\mid}(\sum_{a\in P}a)^n = n!\prod_{i=1}^na_i$$ where $S = \{a_i\mid i\in [n]\}$ which is a multiset of positive integers. It is pretty much similar to the form of principle of inclusion…
Beverlie
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How many arrangements of these letters are there with no pair of consecutive letters the same?

Given $2n$ letters, two of each of $n$ types. How many arrangements of these letters are there with no pair of consecutive letters the same? Please check my solution and tell me where my mistake is! So such arrangements where the pairs of identical…
Johnathan
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Solution for equation with inclusion-exclusion principle.

Using the principle of inclusions and exclusions count how many solutions to the equation $$ x + y + z = 12 $$ $$ 1 \le x \le 5$$ $$ -2 \le y \le 4$$ $$ 0 \le z \le 5 $$ $$ x,y,z \in \mathbb{Z}$$
user180834
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Reverse Inclusion - Exclusion Principle

The Inclusion-Exclusion Principle is usually expressed as a way of determining unions from intersections, i.e. $$\mathbb{P}(A_1\cup A_2)=\mathbb{P}(A_1)+\mathbb{P}(A_2)-\mathbb{P}(A_1\cap A_2)$$ $$\mathbb{P}(A_1\cup A_2\cup…
sheppa28
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Proof make use of Principle of Inclusion and Exclusion

I am trying to prove the equation below with P.I.E : $$\sum_{i=0}^{n}(-1)^n\binom{n}{i}\binom{m+n-i}{k-i} = \binom{m}{k}$$ First RHS is quite simple, i.e., choosing k among m, and then move to LHS, this part is little confusing to me. It represents…
Quanlisp
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Find the value of $\mathop{\sum\sum\sum\sum}_{0\leq i \leq j \leq k \leq l\leq n} 1$

Find the value of $$\mathop{\sum\sum\sum\sum}_{0\leq i \leq j \leq k \leq l\leq n} 1$$ I am not sure but perhaps the answer is ${n+5}\choose 5$. We know that $\displaystyle\mathop{\sum\sum\sum\sum}_{0\leq i < j < k < l\leq n} 1 $ is ${n+1}\choose…
Mathejunior
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How to calculate the number of strings using Inclusion-Exclusion method?

I have difficulties in solving the following problem using the Inclusion-Exclusion method. How many strings of length n >= 3 are there which can be built using A,B,C,D,E , where A occurs at least once, B occurs at least once and C occurs at least…
Teo
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Integers divisible by 4 but not by 3 and 16

For $n \leq 1000$ I am interested in the integers who are divisible by 4 but not by 3 and 16. Say $a_i$ is the property that an integer is divisible by $i$. Inclusion-Exclusion gives us: \begin{align} N(a_{3}'a_{4}a_{16}') = N - N(a_3) - N(a_4') -…
iJup
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inclusion-exclusion problem

How many base-k sequences are there of length n which contain all k possible symbols? I am trying to solve it using inclusion-exclusion. However, I am actually struggling with the part of counting the number of sequences while at least a symbol is…
Chloe
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Inclusion-Exclusion proof for two sets

I know this might sound silly but, it's easy to convince myself that $|A\cup B|=|A|+|B|-|A\cap B|$ but i'm not sure how i would go about proving it. Suppose $A'=A\setminus\{a\}$ and $|A|=n+1$ for some positive integer $n$ then $\begin{align} |A\cup…
pkjag
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is this formula for $N(A \cup B \cup C \cup D)$ correctly applied?

Question : In a school, the students are fans of one or more of the four actors- A,B,C and D.The four actors given in the above order are liked by $230,180,180$ and $220$ students respectively.The no. of students who like exactly $2$ actors for any…
dhruv
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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 $X_{0}$ a subset of $X_{0}\sqcup X_{1}$? I just would…
sheafpants4
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