Hello

The problem is in the picture below.

Any sort of help is very much appreciated :D

Thank you in advance!

https://preview.redd.it/xgcvx7xbex081.png?width=942&format=png&auto=webp&s=ac73f720d4549c7fef687f7349b4f9c448115af8

Recently I took up self-studying probability using Sheldon Ross' A First Course in Probability when I came across some questions that got my friends confused and I didn't know how to explain the concept properly. Namely, when do we use the inclusion-exclusion principle instead of calculating the probability directly. Here is the example that started this question:

https://preview.redd.it/2xbot93xsob61.png?width=1252&format=png&auto=webp&s=8eeb764d5ef396c7810e9743d127501cf773deba

My friend's first instinct was to simply multiply the probabilities that no person gets their hat, something like for person 1: (n-1)/n, for person 2: (n-2)/(n-1), which when all multiplied together:

(n-1)/n * (n-2)/(n-1) * ... * 2/3 * 1/2 = 1/n

However the book offers an alternate solution that when compared for say, n=5, does not yield the same result:

https://preview.redd.it/rl8hatuktob61.png?width=1254&format=png&auto=webp&s=b7a4aecf13fd54f40499a38310c9fb740183bd47

https://preview.redd.it/cp7zvwsktob61.png?width=1257&format=png&auto=webp&s=d376d57f23a51a4029bd03cbceceb8e2f2824000

https://preview.redd.it/w1lsfpsktob61.png?width=1216&format=png&auto=webp&s=bf0fc882640482e5660e05852cfeeae1dd360781

So then my question is, although I understand the intuition and it makes sense, I have no good reason to say why my friend's approach shouldn't be able to work either. Is there a rule of thumb we can adhere to to understand when to use one approach vs the other?

Apply the principle of inclusion and exclusion to find the cardinality of the set

{f|f:{1,2,3,4,5,6} -->{a,b,c,d,e}, f is a surjection}

b.) use Mathematica to generate these functions and check if the numbers are the same.

the --> is an arrow

I am getting -15 people ordered Coffee Eggs and Meat.

80 total people surveyed: 75 ordered coffee, 65 ordered eggs, 45 ordered Meat, 60 ordered Coffee and Eggs, 40 ordered Coffee and Meat, 35 ordered Eggs and Meat.

Calculate number of people who ordered Coffee and Eggs and Meat with Inclusion/Exclusion Principle

Hey guys I'm having a hard time with this question and wrapping my head around it, would love a detailed explanation, thank you!

The question: Count the number of strings of length 9 over the alphabet {a, b, c} subject to each of the following restrictions. -The string has exactly 2 a's or exactly 3 b's. -The string contains at least 8 consecutive identical characters

Thanks for the help!

I am trying to understand the proof for the generalised case of the Inclusion-Exclusion Principle as set out in 9.2.1 and 9.2.2 (Theorem 9.2) of the resource I have linked below.

http://www.maths.manchester.ac.uk/~mrm/Teaching/DiscreteMaths/LectureNotes/InclusionExclusion.pdf

On page 5, where it supposes that x is an element contained in exactly L subsets of the Union of all X sets I don't understand why we need to this. From that point onward I am quite lost and don't understand what is happening.

I am trying to understand what is going on in the proof, however, I can't seem to get my head around what it is trying to say. I have tried using other resources which have either taken an identical or similar approach yet I don't really seem to be making much progress.

Thanks in advance.

#Prompt

Suppose Sue is a Mail Carrier who is crazy. He likes to ensure that none of the n houses on his delivery route get the mail they are supposed to. Your goal, should you choose to accept it, for this sub experience is to use the Principle of Inclusion/Exclusion (PIE). The other method is to use an exponential generating function to solve a recurrence you’ll develop. Put Dn equal to the number of ways Sue can distribute mail to n houses so that none of them gets the correct mail.

#Part One PIE Approach

Use the PIE to determine Dn.

Part Two: A Concise Formula

The Formula you obtain above should involve a truncated power series for e-1. Show Dn = [n!/e + ½], for n > 0 (For n = 0, the formula doesn’t work: D0 = 1 but the formula gives 0.)

#Part Three Exponential Generating Function Approach.

• Prove the Recurrence Dn = (n - 1)Dn-1 + (n-1)Dn-2, for n >= 2 and D0 = 1, D1 = 0

• Deduce, from the above recurrence Dn = nDn-1+(-1)n, for n >= 1, and D0 = 1

• Use an exponential generating function to solve the recurrence from part 1.3.2

I was wondering if someone could walk me through how to solve this. I was sick the day my Prof went over PIE and then when I got back and he talked about it again I didn't really get it. Based on looking stuff up online, I understand that it is basically a way of showing relationships between sets but I don't know how to go about using it to solve this problem. I would appreciate any help.

How many 5-letter words with exactly one vowel, do not contain a B , a C , a D , or a F ?

Workings:

Lwt |B|, |C|, |D|, |F| be the set of numbers that contains the corresponding letter occurring.

Workings:

|U| = C(5,1) 4^21 1^6

|B| = C(5,1) 4^20 1^6 = |C| = |D| = |F|

|BnC| = 2 C(5,1) 4^19 1^6 = ... |DnF|

|BnCnD| = 3 C(5,1) 4^18 1^6 = ... |CnDnF|

|BnCnDnF| = C(5,1) 4!

I believe I am wrong with his info for so long but I have no idea why and what to do to correct this.

How many integers between 1 and 300 are there which are relatively prime to 300? Answer this question using both Euler's phi function and the inclusion Exclusion Principle.

For Euler's: ϕ(n)=n(1 - 1/p1)(1 - 1/p2) ... (1 - 1/pr)

I have 2^2 * 5^2 * 3 = 300.

So, I'd assume 300(1 - 1/2)(1 - 1/5)(1 - 1/3) = 80 would be the answer, but I'm not sure if that's right. Also, I may need some explanation on the Inclusion Exclusion Principle.

Problems are uploaded here: http://imgur.com/a/rXd6Q

Also with at least two pairs of consecutive letters. I understand that for the first part the equation is E2=S2-S3C(3, 1)+S4C(4, 2)-...S11C(11, 9) (Or is it up to S7C(7, 5)?) but how do I find the S terms for this specific problem? Note I am also aware Sn=SUM(N(c1 c2 ... cn)) but how do I apply it to this problem?

Let's say I observe a photon. How would I know that's not, for instance, ten photons in the same state? Once two particles are in the same state, is there some mechanism by which they can diverge?

On my Linux desktops, I feel like I am constantly editing my exclusion files for my home directory backup jobs, as I start using new applications or as applications get updated, more and more crap ends up in my home directory that I don't necessarily need to backup.

Even on a brand new Linux install, things popup in my home directory and I just think "WHAT THE HECK IS THAT?" (like ~/.nv for nvidia, I've never seen until my new laptop)

Then there is stuff that is obviously temporary data (mostly in ~/.cache but you find it all over the place) Seems like there's temporary files all over the place. Then there are also easily replaceable files. (Stuff that can easily be re-downloaded from repositories when you reinstall an application, etc)

I have a mostly manual checklist of things to do when I install a new Linux desktop system. This is because usually, so much has changed since the last time I did a fresh Linux install, that just copying over the home directory isn't feasible when so much changes from version to version of a Linux distro. (Not like Windows is any better in this regard)

So this got me thinking. Is this something that should probably become a new standard. Every application can put a file somewhere that defines what files are important and which are not. Perhaps they can even include a script to determine what files are important and which files are not.

Perhaps they can even categorize them into folders such as:

• CRITICALLY-IMPORTANT: Such as your bitcoin wallet files. Don't want to forget that.
• TROUBLESHOOTING-LOGFILES: Probably safe to delete, not important to backup.
• SECURITY-LOGFILES: Probably don't NEED to back them up, but you might want to.
• SAFE-TO-DELETE: List of files that you can delete without hesitation, and definitely don't need to backup. (such as .thumbnails, cache, etc) This would be nice just if you need to clear up some disk space.
• DO-NOT-BACKUP: Sometimes restoring data can be MORE problematic. For example, maybe you have some kind of highly available system, but restoring old data can cause problems.
• DATA-ONLY-BACKUP: Files that ONLY contain data that would need to be restored, but not files necessary for the application to run (such as libraries, etc).

Then within those folders, each application can put scripts such as "firefox" "gnome" etc.

The possibilities and flexibility are endless.These folders can be stored in places like /etc/backup and ~/backup or ~/.backup or someth

Hi, I have to solve this combinatorics problem and I think this is the best forum where to post.

The problem is: There are 4 children who dislike their toys, in how many ways can the children exchange their toys without ending up with their own toy in the hands? I know I have to use the principle of inclusion and exclusion but I haven't managed to find a solution. Thank you

Is it verifiable by experiments or is it a way to explain what happens given no other explaination is present, this isn't meant is a rude way. I genuinely don't understand

https://en.m.wikipedia.org/wiki/Exchange_interaction

One of my teachers asked the question and I wanna know how it is related in the atomic level. Answers are much appreciated.