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.

Filter by
Sorted by
Tagged with
0
votes
1answer
49 views

Solving the same problem in different ways obtaining different answers

I am trying to solve this question in different ways, but results don't tally, and some of them look incorrect to me. Can someone point out where are the loopholes in these logics? Count the number ...
1
vote
1answer
21 views

Is this Inclusion-Exclusion: $m$ balls colored $n$ colors such that each color is represented?

Suppose you have $m$ balls each of which can be colored by $n<m$ different colors, and your paint cans are unlimited. I want to count the number of ways that you can color balls such that each ...
1
vote
1answer
35 views

Generalization of inclusion-exclusion rule

This is taken from Pollard, A User's Guide to Measure Theoretic Probability problem 1.1. Let $A_1, A_N$ be events in some probability space. Denote $\cap_{i \in J}A_i$ as $A_J$ for some set $J \...
1
vote
3answers
49 views

In how many ways can we split 8 animals into four groups with these requirements…

So we have $2$ cats, $2$ dogs, $2$ lions, $2$ tigers, and we want to split them into $4$ groups such that each group has $2$ none similar animals. In how many ways we can do that? I'm trying to use ...
2
votes
1answer
43 views

Principle of inclusion exclusion and counting surjective functions

Let $f$ be a function from $A \rightarrow B$. $|A| = 4, |B| = 3$ The number of surjective functions by applying the principle of inclusion exclusion is given by: $3^{4} - \binom{3}{1} 2^{4} + \binom{3}...
0
votes
1answer
55 views

Probability of no matches taking $N$ cards out of $rN$ cards at random and matching them against a target deck

In Feller's introduction to probability the answer to this problem is given by: $ $$\tag1 p = \sum_{v={2}}^{N} (-1)^{v}{N \choose v}r^{v}{\frac{(rN-v)!}{rN!}}$ Acording to the means of the chapter ...
0
votes
0answers
30 views

How is the inclusion map both an Immersion and Submersion between Manifolds

Hi i am trying to figure out why an inclusion map is both and immersion and submersion. This is what i have tried so far.Let $S$ be an open subset of a manifold $M$. Now the inclusion mapping is $\...
1
vote
2answers
76 views

Evaluating $\sum_{i=0}^{30}(-1)^i {{30} \choose {i}}(30-i)^{31}$

As the title says, I've been asked to evaluate this sum: $$\sum_{i=0}^{30}(-1)^i {{30} \choose {i}}(30-i)^{31}$$ and have been unable to do so. The possible answers are $$\frac {30\cdot31!} 2, \frac {...
1
vote
1answer
47 views

Finding the least number of plants watered by three gardeners

There are 100 plants and three gardeners A, B, and C. If A watered 68 plants, B watered 78 plants, and C watered 88 plants, at least how many plants are watered by all three gardeners? I drew a Venn ...
0
votes
1answer
24 views

The Principle of inclusion and exclusion to find probability [closed]

A 5-card hard is dealt from a standard deck of 52 cards. Find the Probability of at least 1 heart and 1 spade is among the 5 cards, Using Inclusion and exclusion. So far I have 52C5-2*47C5 (47C5 from ...
1
vote
1answer
59 views

How many ways can we write a word of 4 letters from the group of {1,2,3,4} without the sequence 12 and 23?

How many ways can we write a word of 4 letters from the group of {1,2,3,4} without sequence 12 and 23? the options are: 16 2.256 3.172 4.24 I think we can repeat on the same letter so I tried to ...
1
vote
0answers
30 views

Finding the probability of occurence of at least $m$ and exactly $m$ events out of $(A_i)_1^n$

Let there be $n$ events given by $A_1,A_2,\ldots,A_n \subseteq \Omega$. $B_m$ denotes the event wherein at least $m$ events occur, and $C_m$ denotes the event wherein exactly $m$ events occur. I must ...
0
votes
1answer
29 views

Question about when to use inclusion-exclusion principle.

I was solving a question that says that we're spreading $N^2$ balls into a grid $N \times N$, that has $N^2$ $1\times 1$ squares. I had these two questions one after another and I failed the second ...
1
vote
1answer
53 views

How does inclusion exclusion relate to the complements

Let $A_1,A_2,A_3,A_4$ be events such that for every $i,j,k=1,2,3,4$, $P(A_i) = \frac{1}{2}$, $P(A_i \cap A_j)= \frac{1}{3},\quad i\ne j$, $P(A_i\cap A_j\cap A_k) = \frac{1}{4},\quad i\ne j, j\ne k, k\...
1
vote
0answers
31 views

Is there any way to write inclusion exclusion principle recursively? [duplicate]

I need to use inclusion exclusion principle to calculate the probability of several random events, but the problem is that the computational complexity of this equation is exponential. So I need a ...
1
vote
1answer
47 views

Recursive inclusion exclusion principle

The inclusion exclusion principle is as follows: $P\Bigl(\bigcup_{i=1}^n E_i\Bigr) = \sum_{i\le n} P(E_i) - \sum_{i_1<i_2}P(E_{i_1}\cap E_{i_2})+\sum_{i_1<i_2<i_3}P(E_{i_1}\cap E_{i_2}\cap E_{...
1
vote
0answers
13 views

Approximate the inclusion exclusion principle

Suppose the following relation is established: $P\Bigl(A\cup B \cup C\cup D\Bigr) < P\Bigl(E\cup F\cup G\Bigr)$ based on boole's inequality, for each of the above probabilities we can have the ...
5
votes
2answers
70 views

Combinatorics - letters forbidden sequence.

I am trying to solve a question. I have 15 pens, where 5 are colored in Red, 5 in Green, and 5 in Blue. How many combinations can I order these pens so that there is no sequence of 5 pens with the ...
0
votes
0answers
14 views

Lower bounds of the inclusion exclusion principle

Suppose the following relation is established: $P\Bigl(A\cup B \cup C\cup D\Bigr) < P\Bigl(E\cup F\cup G\Bigr)$ based on boole's inequality, for each of the above probabilities we can have the ...
0
votes
2answers
54 views

How Many Positive Integers Below $201$ Are Multiples of $4$ But Not Multiples of $10$?

I have recently been starting to work on counting problems, and currently, I am working on this one, but cannot figure it out: How many positive integers less than 201 are multiples of 4 but not ...
0
votes
0answers
13 views

A formula connected with existence of big intersections of given sets.

Assume that we have following sets $A_1,A_2,...,A_s\subseteq\{1,2,...,t\}$ Also we have two natural numbers $k,l$, such $k\le s$ and $l \le t$ Let there be a function $f$, such: $$f(k,l;s,t)=\min\{\...
1
vote
0answers
44 views

Different ways to prepare 5 dishes

There are 3 entities of ingredient A, 8 of ingredient B, 12 of C and 7 of D. Person $P_3$ may consume his dish iif it contains an even number of ingredients. Person $P_4$ requires that his dish ...
3
votes
2answers
77 views

How many permutations of numbers $(1,2,3, .., n)$ are there in which at least $2$ elements are in their original place?

How many permutations of numbers $$(1,2,3, .., n)$$ are there in which at least $2$ elements are in their original place? My idea for solving: $n! - |\text{no element stands in its original place}| - |...
1
vote
1answer
26 views

Proof for $I \in \binom {[r]}{k}$ with $k \in \mathbb{N_0}$ there exist exactly $(r-k)^n$ r-tuples $(X_1, …, X_r)$

For $n, r \in \mathbb{N}$, $(X_1, ...,X_r)$ is an ordered partition of $[n]$ in $r$ non-empty sets, if $X_1,..., X_r$ are disjoint and non-empty subsets of $[n]$ for which it holds that $X_1 \cup .....
0
votes
0answers
17 views

Proof of $S_{n,r} = \frac{1}{r!} \sum_{k=0}^r (-1)^{r-k} \binom{r}{k} k^n$ with inclusion-exclusion principle [duplicate]

For $n, r \in \mathbb{N}$, $(X_1, ...,X_r)$ is an ordered partition of $[n]$ in $r$ non-empty sets, if $X_1,..., X_r$ are disjoint and non-empty subsets of $[n]$ for which it holds that $X_1 \cup .....
1
vote
1answer
39 views

Divide $n > 10$ distinct elements $\{1,2, \dots, n\}$ to $3$ distinct sets $A,B,C$ where $|A|=5$ and $|B|>|C|$

I am trying to use the inclusion-exclusion principle here but having a problem calculating the cases. Assume $D_1$ - the possible ways to choose a number not equal to $5$ members for $A$ and $D_2$ - ...
5
votes
3answers
89 views

How many ways are there of creating an $8$ character password with a digit, a lowercase letter, and $2$ capital letters?

A valid password for the bank's website consists of 8 characters (digits and letters in English) and must contain at least one digit, at least one small English letter, and at least two capital ...
0
votes
3answers
109 views

How many integers between $1$ and $1000$ are divisible by $2, 4,$ or $7?$

How many integers between 1 and 1000 are divisible by $2, 4,$ or $7?$ $A \cup B = A+B- A \cap B=500+142-71=571$
0
votes
0answers
36 views

NIMO April 2017 conditional probability / PIE system

Two neighboring towns, MWMTown and NIMOTown, have a strange relationship with regard to weather. On a certain day, the probability that it is sunny in either town is $\frac{1}{23}$ greater than the ...
0
votes
0answers
24 views

How to denote an equal frequency of common subsequences in two sequences?

What would be the most appropriate symbol for denoting an equal number of subsequences in two sequences? For example: $$A = (1,2,3,4),\ B=(9,2,3,1,2,3,4,1,2,3,4),\ C = (9,1,2,6,1,2,3,4,5,6,7).$$ Let $\...
0
votes
1answer
36 views

Can someone help me with this sets question? (How many students knew no coding language?)

Q: In a survey of a group of 100 computing students, it was found that 32 knew Java, 33 knew C, 32 knew Python, 5 knew both Java and C, 7 knew both C and Python while none of the students knew both ...
0
votes
1answer
82 views

Inclusion–exclusion principle, password combinations

Co-workers are required to create 6-character long passwords. The letters must be from lowercase letters or digits. Each password must start with a lowercase letter and end with a digit and contain at ...
0
votes
0answers
40 views

Find rook polynomial for a full $n\times n$ board

Alan Tucker's Applied Combinatorics question. I have a full $n \times n$ board and I need to find the rook polynomial. I am not sure if I comprehend the method wrong, but I am approaching this ...
1
vote
1answer
46 views

Proof of Inclusion Exclusion Principle

Show that $|A \cup B| + |A \cap B|= |A| + |B|$ for two finite sets $A$ and $B$. Can you please check the proof below, and let me know if it's right? It makes me a bit uneasy for some reason, and I can'...
0
votes
0answers
36 views

Finding number of arrays such that first and last elements are same with unequal adjacent elements.

The first and last elements of the array are the same. The size of the array is n and the array element can have values in [1, m]...
1
vote
2answers
124 views

How many permutations of the string “000011112222” contain the substring "2020”?

So far, this is what I know. There are 9 ways to place 2020 and $\frac{8!}{4!2!2!}$ ways to arrange the remaining numbers. Im having a problem tackling the overcounting case.. If there are 2 seprate ...
2
votes
4answers
67 views

How many different $ 8$ bit strings begin with $000$, end with $001$, or have $10$ as the $4th$ and $5th$ bits, respectively?

I keep obtaining $127$ as my final answer yet I know this is incorrect. I start with the # of combinations for $8$ bit strings that begin with $000$, which is $2^5$. Then calculate the # of $8$ bit ...
0
votes
1answer
18 views

Inclusion-exclusion of values of $\Omega$ (the arithmetic function).

For any number $n \in \Bbb{Z}$, define $$ f(n) = \Omega(n) - \sum_{q \mid n} \Omega(n/q) + \sum_{q,q' \mid n}\Omega(n/(qq')) - \dots \pm \Omega(1),$$ where each sum is taken over only prime divisors ...
1
vote
2answers
30 views

When can you not use inclusion-exclusion in probability?

I came across this question: Alice and Bob both try to climb a rope. Alice and Bob will get to the top with probability 1/3 and 1/4 respectively. given that exactly one person got to the top, what is ...
0
votes
1answer
32 views

No-Adjacent objects using inclusion exclusion

Teacher X, Y, Z and 6 students sitting in a row. How many ways can the teachers choose their seats so that there are no 2 adjacent teachers? I know that you can solve it by: (s) = student. We put 1 ...
0
votes
0answers
31 views

Q: Let |A|=n,|B|=m be finite sets then (Using Inclusion-Exclusion Law)

due to question above we need to answer |A∪B|, |A∩B| I dont know how solve this question with value of just A and B this is rules |A∪B|= |A|+|B - |A∩B| |A∩B|= |A|+|B| -|A∪B|
1
vote
1answer
34 views

A problem on Principle of inclusion and exclusion.

I am having a little bit of trouble understanding this part in my book. I know the normal PIE that is n(A+B) = n(A) + n(B) - n(AB) and then so on for more sets but I am not able to understand this ...
1
vote
1answer
19 views

Probability question related to independent/dependent event and conditional probability

An investment company estimates that the probability that stocks a and b go up is 61 % and 75%, respectively. Let A be the event that stock a goes up, and let B be the event that stock b goes up. ...
-1
votes
1answer
52 views

Number of strings containing each vowel at least once?

Consider the English alphabet {$a,b,c,...,z$}. How many strings of length $n$ can be formed in which each vowel {$a,e,i,o,u$} appears at least once?
2
votes
1answer
89 views

How many solutions does the equation have? (Inclusion-exclusion principle)

How many solutions does the equation $x_1+x_2+x_3+x_4=1$ have? $x_i$ are integers between $-3$ and $3$. Hints please!
1
vote
2answers
69 views

How many positive integers $\le 462$ are relatively prime to $462$?

How many positive integers less than or equal to $462$ are relatively prime to $462$? My Approach Well my approach is basically that we find $|A \cup B|$ where: $A = \{\text{all prime numbers} \le ...
2
votes
1answer
38 views

Question on combinatorics and the inclusion-exclusion principle

Find the number of n digit-numbers formed using the first 5 natural numbers, that contain the digits '2' and '4', essentially. I tried attempting this with the inclusion- exclusion principle but got ...
2
votes
0answers
117 views

What is the expected number of consecutive digit pairs “23” in a random integer between 1 and 1,000,000?

had this question on a test and wasn't sure whether my solution is correct. We can use linearity of expectation to look at each two consecutive digits and their expectation for digits 23 as a pair (...
2
votes
0answers
73 views

How many numbers between $1$ and $200$ are multiples of $3$ or $5$ or $7$?

How many numbers between $1$ and $200$ are multiples of $3$ or $5$ or $7$? (a) $107$ (b) $108$ (c) $109$ (d) $110$ (e) $111$ I keep getting $108$ as my answer using PIE, and getting $66+40+28-13-5-9+1=...
0
votes
1answer
44 views

If L1 is decidable and if L2 is included in L1, is L2 also decidable?

Is this statement true or false and why ? If $L_{1}$ is decidable and if $L_{2}$$\subseteq$$L_{1}$ then $L_{2}$ is also decidable. I would be tempted to say yes, but i am really not sure. I am also ...

1
2 3 4 5
24