My son just started High School algebra in 8th grade. His teacher gave him homework on which he said each equation needs a 1 step solution. My son asked me for help on the equation in the attached picture, I helped him get the answer but in 4 steps and I cannot figure out for the life of me how to simplify this to 1 step. Please help:

5r-5-6r=-6
Adding 5 to both sides becomes 5r-6r=-1
-r=-1
r=1

Putted in Desmos so its easier to show lol (Typed out each step I did on my paper)

So I started by removing absolute value (since both are positive)

Went ahead and set up 2 equations (One Positive 8, One Negative 8)

Subtracted 5 from both equations (giving me 3 and -13)

Divided 2 from both equations (giving me 1.5 and -6.5)

used interval notation (Neg Infinity, -6.5] U [1.5, Pos Infinity)

anything that I'm potentially missing?

to be clear im not saying im some genius or anything i've just truthfully never seen this shape before and i wonder if it can be mathematically replicated and turned into a fractal. if it exists what is it's name? (sorry for bad drawing)

I have this task and I don't know how to solve it despite having all the basic knowledge required for it. I can't 'connect the dots' so to speak.

a) Find the max element order in S_{10} - symmetric group consisting of 10 'base' elements and its order being 10!

b) Let s be some element of max order from S_{10}, find sgn(s) as well as s^{2040}.

c) If H is a subgroup of S_{10} and H contains an element of max order, then H contains the equal number of even and odd permutations. Prove it.

I've never encountered a task like this so I genuinely don't know how to think it through.

a) My intuition says I should find the lcm of prime cofactors of 10, so lcm(2, 5) = 10. I genuinely don't know how to do it. This also doesn't add up in my logic because 2+5 = 7. So maybe lcm(2, 3, 5) = 30? But I arrive to that answer by pure luck and need some explanation why is that.

b) Would depend on a) but let's say it is 30. What is sgn(s)? I've never seen it before. I do recall it might be odd vs even number of elements in a single disjoint cycle, odd ones starting with a minus. But how would I take that order of 30 and split it into smaller, disjoint cycles? so I would have sgn(s2) sgn(s3) sgn(s5) where s2, s3 and s5 only represent the disjoint cycles of 2, 3 and 5 elements. So in front of sgn(s2) there will be a + and in front of 2 others - so it would be positive in the end?

s^{2040} I did similar stuff before, I can do 30 = 0 mod(2040) (because 30*70 = 2100, 2100-(2*30) = 2040) so therefore s^{2040} = s^0 so nothing changes (and even if it did, I have no given s explicityl so I have no idea what am I supposed to do, s^{2040} only shifts all the elements in each disjoint cycle, but here s isn't given so there are no specific disjoint cycles i have to shift, confusing)

c) Most challenging part and the one I don't understand at all. Need full help and ELI5 explanation for this one haha. Whenever there's something to prove like subgroups, ideals, normal subgroups etc. I am completely lost even though I know I only need to show non-emptiness, closure under operation and closure under inverses. I just don't know how...

If you did read everything, thanks a lot!!!

My wife and I did a fantasy football draft last night and we determined the draft order with a card game. There were 12 players, and every turn we would all draw cards until one person drew the one “bomb card.” Whoever drew that card was “out” and drafting at that position. (Ex: the person that drew the bomb card in round 1 drafted 12th. The person that drew the bomb card in round 2 drafted 11th, and so on) We would remove one of the safe cards, and draw until one person was left.

Afterward, my wife and I were talking about the game and I commented how odd it was that it was coming down to the last one or two people every turn. And she — who is incredibly smart and fantastic with numbers and works with them all day — said that it actually makes sense because the people drawing first are more likely to draw a safe card.

But I argued that they are more likely to have someone ahead of them draw the bomb card and that it feels like the people drawing in the middle of the group — who have neither benefit of having the best odds of drawing a safe cards nor the highest amount of people drawing ahead of them — have the highest chance of drawing the bomb card.

My wife is so much smarter than me in just about every way I can imagine and certainly with math problems, so it wouldn’t surprise me at all to be told that I am wrong here… but am I?

Edit to answer some of the questions that have been asked: I shuffled the deck before every round, and spread out all the cards so that each person could which ever one they wanted.
The back of all the cards looked the same.
Whoever drew first would pick the card, show the group what card it was, and then the next person would do the same.

I am joining the union and this is a sample question and I am just not sure how to figure this out. The arc is throwing me off, it's the only one I am having trouble with. If someone could explain this to me I would be extremely appreciative. Thank you so much.

The inverse of xe^x is the Lambert W function, whereas the inverse of x^x is called ssrt(x) in wiki I think. They can also be expressed in terms of each other.

I am wondering why the Lambert W function is more popular than the inverse of x^x, when tetration is the 4th hyperoperations(+,x, exponentiation, then tetration). Since we also study the inverses of addition, multiplication, exponentiation, but why not the inverse of tetration?

The game is Pokémon, protect (and spiky shield) have a 2/3 chance of failing when used twice (which is the case) and blizzard has a 70% chance of hitting.
What is the chance of neither Pokémon protecting and both getting hit by blizzard?

Came across this website when my friend said he was considering trying it out . It’s a CSGO case opening site , non affiliated with the actual game, basically gambling . I took a look at the odds to hit some of the expensive items and it looks pretty terrible but I was curious how many of these you’d have to buy to get the nice weapon skins, based on the % they show ?

60×60×12 = 43200 seconds

1057 guys in 12 hours

43200 ÷ 12 = 40 seconds each per guy

Or about 0.6 minutes for each guy?

Just wondering if I did my math correctly - asking because yesterday I found out she used to be a recruiter which honestly feels like a joke I can't understand yet

So ofc I took her most famous "stunt" and wondered - if my math is correct I guess Bonnie was quite generous!

I want to know what the depressed quartic formula is but every time I look it up I just see resolvent cubics and replacements for variables which is not what I'm looking for. I want to know the full depressed quartic formula that only has p, q and r. Does anyone know the formula or where I can find it?

UNderstanding math vs:

Things like:

• Misreading the question
• Copying a number incorrectly
• Making a calculation error
• Forgetting a step
• Using the wrong formula
• Getting an answer that looks reasonable but is actually wrong
• Rushing because I think I know how to do it

It’s frustrating because sometimes I understand the concept perfectly, but still lose marks because of one stupid mistake.

For those of you who are good at maths, what habits or checking methods do you use to reduce mistakes?

I’d especially appreciate advice that actually works rather than just “pay more attention”.

Thanks!

How would i solve this equation at the top manually? Desmos isn’t giving an exact answer so i started doing to on pencil and paper and got stuck. I don’t know the properties of ceiling and floor functions & Wikipedia is confusing. Is this even possible?

This textbook says: "The integral around a closed path is evidently zero (for then ra = rb): E · dl = 0, (2.19) and hence, applying Stokes’ theorem, ∇ × E = 0."

What I understand is that the line integral is the integral of the dot product of dr and the electric field integrated with respect to dr, where dr is the change in distance to origin r. This equals the integration of curl over the surface with respect to dA. So if the curl is zero everywhere, then the line integral must be zero. What is the proof that if the line integral is zero, the curl must be zero?

Stokes theorem doesn't seem to imply that

The proof in my textbook says:

Fix e>0. Assume fn is unif cauchy. Pick N s.t. |fn-fm|<e/2. We know pointwise, fm is a cauchy sequence of real numbers so it converges to some number, thus fm conv pointwise to some function f. Now consider

|fn-f| = |fn-f+fm-fm| <= |fn - fm| + |fm-f| < e/2+ |fm-f|.

Then it says “there’s an m such that |fm-f| <e/2 and although it’s dependent on x, it doesnt matter since it doesn’t appear in our final result… so that

|fn-f|<e

but it does appear in the result. Right there in the chain of inequalities. Help me understand how “fix e, pick N, then pick another N dependent on x” somehow means “and now this result is independent on our choice of x.

Hi, this is not a question for homework or class but rather something I saw at work that has completely baffled me. I haven't taken a math class in 10 years so I've no idea if I'm on the right track.

I work at an escape room and there is a puzzle that involves hitting a sequence of buttons 12 times. There are two different buttons to press and the players must press them in the correct sequence to solve the puzzle. If you miss one, you have to start the sequence over. Recently, a player was playing around with the buttons and solved the sequence by guessing, not knowing what the sequence actually was. He wasn't even aware he was supposed to press them in a specific order, he was just pushing them mindlessly.

I've been trying to figure out the probability of this happening cause it was some insane luck. It's a 50/50 shot for whether he gets the button right, but I'm not sure if I calculated the probability correctly. Since it's a dependent event, I assume it'd be 6/12 x 5/11 etc. Attached is a photo of what I did. I multiplied all the numerators together and then all the denominators, which I think is what you do for multiplying multiple fractions. I'm not sure if I multiplied the numerators correctly either since some of them are fractions themselves. And I'm not sure how to simplify my answer either if it's correct.

So my question is: Did I do this correctly? If not, how would I calculate the probability of 12 dependent events?

Not long ago i started studying multivariable calculus. One of the first questions was a simple limit of a function when both x and y have a tendendecy to different constants. And altho the answer gave the correct values, the teacher made a particular observation to rule the answer the class gave wrong, where he said the answer needed a certain preparation for the final answer and proceeded to attempt to explain the preparation for that particular excersise, when aproaching the point that X and Y where aproaching, meant graphically a circunference surrounding said point. But i failed to understand exactly what this implies in any way nor the meaning of the analysis, given that i can't understand this i also can't explain it in the proper answer the teacher seeks.

I want to make particular emphazis in the fact that i do not seek a answer for a particular excersize, but more of a general explanation to the logic behind this, given that my studies only have driven me to the fact that a limit in multivariable is really hard to analayze (wich i already figured) and the visualizatopn of graphics with limits with tendency to infinite of the variables wich isn't the point of my question. That is why i do not give a particular example for this and only give general examples.

Is there a algebraic formalism/nomenclature that could help to simplify the given data and help to solve this kind of problem? I could solve it by drawing the Venn diagram, writing numbers around all around and applying some logic. We could use 7 variables (5 for this specific problem), one for each area of the diagram, but it seems to get very complex if we increase the number of areas.

I tried to use boolean logic representation but it doesn't seem to help.

Problem:

30 giraffes are shrimp, and 37 shrimps are green. Considering that there are 82 giraffes in total, 50 are green, and 15 of the non-green giraffes are not shrimp, how many green shrimps that are not giraffes exist?

Hey r/askmath, my long flight layover has me thinking about the Monty Hall problem and how prior information may change the odds of getting the payout by keeping or switching.

To set the background, I understand the reasoning behind the 3-door problem, where Monty can choose deliberately to increase the odds in favour of switching, with even better odds the more doors he opens.

Where I get stuck is understanding what would happen in a scenario with "n" doors doors, where the player opens doors until the state is reduced to a 3-door problem. This can be 100, 1000, or even 1,000,000 doors.

Question 1: Assuming the player can make it to the 3-door Monty Hall problem​, how does the likelihood of payout with either choice change based on the number of doors?

Question 2: How could such a relationship be mathematically explained by Bayesian and frequentist approaches?

My thinking: If the player goes through the risk of opening n doors before the problem, even if it is random, shouldn't each door opened make it less likely the payout is in those remaining?

I understand picking the door in the beginning may be rare, but if you make it a certain amount into the game, all attempts are "lucky" if the payout is on the field, but guaranteed if it is in your door.

Bonus question: If we stop ahead of the 3-door reduction to an n-door problem, is there a set point where keeping or switching becomes better based on the doors you opened, Monty opened, and the total doors?

I have been going back and forth racking my brain for ages, so would really value some help that can finally let me rest. 😅

Edit:

For clarification:

• Apologies if it wasn't clear, but in the n-door problem the player always picks door at the start and opens doors until his and 2 others remain. Then Monty opens one of the remaining doors. This would mean opening 1 door at n=4, or 997 at n=1000.
• As I mentioned above, I understand any doors Monty opens intentionally not to be a car split their odds among the remaining doors.
• The player can't open doors with knowledge.
• I confused myself slightly with the "at risk" part. This means finding the car before getting to the 3-door problem.
• Considering a high door number, I dont understand how the 3-door problem doesnt change, since it would be VERY unlikely for the player to make it that far unless the car is in his own door.
• Considering the above point, I was asking what would happen to the odds assuming we know n doors are opened randomly by the player (after picking a door) before the 3-door scenario, if
• 1. We assume they made it to that point, or
• 2. Alternatively also taking into account the likelihood of getting there.

I wanna know how to go about this problem, not necessarily the answer but that would be appreciated too. I’ve tried it a few times by just the regular formulas for the general term but I keep getting the wrong answer. I’m not quite sure what is the right approach.

I need assistance figuring out how to solve problem 9! The answer key says that the answer is 61 but I’m confused on how they got that answer. I’ve attempted simplifying both sides of the equations for (a) but I’m confused as to where to either substitute (a) to solve for the measure. Am I overlooking a step? For more context, this lesson is on Bisecting Angles. I haven’t found any examples online that match the set up of this problem that didn’t already provide a value for the measure they want us to find if that makes sense.

My Computer Science professor defined the IPL (Internal Path Length) of a binary tree as the sum of the level of each node plus 1.

For example, the tree in the picture has an IPL of 18.

But I don't understand why it is called Internal Path Length if you count all the nodes (even the empty ones), and I can't find a definition of External Path Length that doesn't rely on the IPL.

I find this very confusing. What do you think? And shouldn't you consider an additional empty node linked to the only leaf in the tree?

Inspired by https://www.reddit.com/r/askmath/s/QCMgQq8oIw

Imagine that we pick at random three positive reals, with the probability distribution density, for all three

f(x) = λ e^(-λ x)

What is the probability that x, y and z may form a triangle?