Thank you so much, Steven! I'm glad that we could get a chuckle out of somebody. Sometimes, it's hard to laugh when faced with an endless pile of GMAT videos. :) Have fun studying, and keep us posted on your progress!

This absolutely made my day. Thank you so much for the kind words! Have fun studying, and please keep us posted on your progress -- we always like good news. :) - Charles

Awesome, I'm glad that it was helpful, and thank you so much for the kind words. Have fun studying!

This made my day -- thank you so much for the kind words. Have fun studying!

these Qs are brutalll😅 but fun!

I love this. Those are two really tough questions, so I think that's a sign that your intuition might be pretty darned good. Have fun studying, and please keep us posted on your progress!

Thank you, Kanika! Have fun studying.

Thank you so much, and have fun studying!

We’re not spreading out. The first counting is the number of places which serve 1 item. The second counting is that of the number of places that serve 2 items. Similarly the third counting is of the number of places that serve 3 items. Since we have to minimise the third counting, we maximise the other 2 and that’s why they are set as 120. Hope that helps!

It is possible for some locations to introduce none of the three items. Consider the scenario in which 72 locations introduced all three items, another 13 introduced item X and Y only, and a further 5 introduced item X only. In this scenario, there are 30 locations that didn't introduce any of the three items. I hoe that helps!

Thank you for the question, Himanshu! The data I cited in the video was from GMAT Club's autopsy of the two free, official mba.com practice tests. The data is available at these links: - gmatclub.com/forum/what-is-tested-on-the-gmat-344670.html#p2698641 - gmatclub.com/forum/hardest-verbal-questions-tested-on-the-gmat-prep-software-345076.html Take these with a HUGE grain of salt though, especially for verbal. GMAT and EA questions often span multiple topics, and while it's certainly nice to have a general sense of which topics are truly important, you don't want to take the numbers TOO literally. That's particularly true on sentence correction. Very, very few SC questions test just one concept. Meaning and logic are part of almost every question, and several different grammar/usage concepts often appear in any individual question. So while I enjoy the data that GMAT Club compiled, I'm not sure that it's terribly actionable for SC in particular. I hope that helps a bit!

You're welcome! We're so pleased you're enjoying the series. Unfortunately, you're asking the $64,000 question here and the disappointing answer is that it depends entirely on the question. The headings in the table are designed to be as general as possible, so while you might have to change "people to something else in a different question, "countings" can be applied to any question. This means you should be able to build a table that looks similar, if not identical, in structure to the one demonstrated here. However, the thinking behind linking "countings" to "the number of things ahead of beach holidays" is the challenge in this question. Hopefully having seen how Charles does it here, you'll be in a better position to do it next time you see a question like this. I'm sorry that's not a brilliant answer but if it's any consolation, the chances of you seeing a question like this are very slim. I hope that helps a little bit, but please let us know if you have any follow-up questions!

Close! Q6 on the video doesn't say anything at all about whether the events are dependent or independent -- so based on the phrasing alone, it's possible that the two events are independent. Technically speaking, you don't need to assume that the events are dependent. Of course, if the events were actually independent, then the probability that both events occur would just be the product of the individual probabilities (0.7 * 0.85 = 0.595). In this case, that's not the correct answer, as explained in the video. So it turns out that the two events are dependent, not independent. But you don't need to assume that the events are dependent in the beginning -- you just need to recognize that the events COULD be dependent, and then reason through the question from there. The same is almost certainly true of the question you saw on GMAT Club. If the answer was the lower of the two probabilities, then the events were almost certainly dependent. Maybe the question didn't explicitly state any assumptions, but it probably doesn't need to -- generally speaking, it will be clear from the context whether the events are independent or dependent. Or in some cases (like Q6), it will be clear that either is fair game. Also, the GMAT doesn't really require you to have a perfect, technical understanding of independence vs. dependence. If you're familiar with those concepts, that's great. But it's not really a core part of the exam. I hope that helps a bit!

This is the sort of question that's hard to explain in the limited formatting of a KZfaq comment, but I'll do my best. In those 3-way overlapping sets questions, the core idea is that there's some number of actual people, but if we add up the number of "countings" (in your example, the total number of roster spots -- so 26 + 28 + 18 = 72), we'll get a larger number. On these questions, our job is always to disentangle the number of "overcountings". With that general idea in mind, the chart will still work, even though we have different pieces of missing information. You can use the same columns as we do in, say, Q3. The first column is the number of "countings" (classes taken) per person; here, we have people who are in 1, 2, or 3 different classes. The second column is the number of people with that number of classes. And the third column is the "countings" -- so the product of the first two columns. Now, we know that the total "countings" = 72, so that's the sum of the numbers in the third column. We also know that 4 names are on all 3 rosters. And here's the tricky part: how many people are on exactly 2 rosters? Well, 4 people are on all 3 rosters, so 5 are in E & M (but not S), 3 are on E & S (but not M), and 6 are on M & S (but not E). So we have 13 people on exactly 2 rosters. We're almost done now! Let x = the number of people on exactly 1 class roster. Then, all you have to do is multiply across, and you'll be able to solve column 3 for x. And the total number of people will be x + 13 + 4. I hope that helps a bit!

Very different perspective and amazingly out of box. Thanks for sharing.

Genius

Yeah same question @GMATNinjaTutoring

Hey! How did your GMAT Go?

First of all, forget ur way of solving this question so that ur way of doing doesn't mix up with my way in order to understand the way I did this 😂 & please give proper gaps at the places where I have inserted *comma* to understand this solution in the way I perceived it. 90,85 & 72 introduced but in total there are 120 franchises(locations) which implies only one item, can be introduced, at 120 locations maximum. Similarly, maximum 120 franchises can introduce exactly 2 items and then we are left with 7 items which can be introduced at 7 franchises. We did so because we wanted to minimize exactly 3 items' locations. In order to do that we'll have to maximize the remaining (exactly 1 & exactly 2). So, we did in this way. Hope this helps and if u r having still having any doubt, plz reply back.

Yeah, I agree with your input as well. The approach doesn't make sense if you look at it this way. Could GMAT Ninja offer some clarification?

There are always multiple ways to approach any individual GMAT question, so it's certainly possible to use a different sort of algebra to arrive at the correct answer. The answer presented in the video is definitely correct, and we happened to get there using a method that works well for us and for many of our students. If that particular way of thinking doesn't resonate with you, no worries -- there are other ways to answer the question that are perfectly effective, if potentially a bit less efficient. If you arrived at a different answer using algebra, it's probably because there was a mistake somewhere in your equations or calculations. I can't guess what, exactly, your mistake was, though. I hope that helps a bit!

If you take one of the 3-way questions in the video and change it so that people can do NONE of the activities, the question wouldn't be solvable at all. In theory, you could still organize the information using the method in the video, but you'll end up with too many unknowns -- if you're really curious, try it out, and watch what happens. Basically, you'll end up with three variables, and you'll have a mess on your hands. In other words: you won't see a problem quite like that on the GMAT, where you could have people doing 0, 1, 2, or 3 of the activities. So don't lose too much sleep over it. I hope that helps!

@@GMATNinjaTutoring Thank you for the quick response. I've come across questions in which they provide the neither in the option and then they ask for 2's only. So it's not an additional variable but an additional number so should be solvable right? Also, the way I'm thinking about it is for 'neither option' wouldn't the left most column be 0 in the above method which would lead to (incorrectly so) as the '# of counting as = 0 multiplied by # of people in neither = 0 In addition to this, could the method in the video be used if we are not given the number of people who have chosen 2 options only (exclusive of people who have chosen 3) i.e we are given people who have done two tasks (which includes people who could have also done 3 of the tasks)? Again, thanks for taking the time out for these replies. Regards

@@someshshahiSure, it's totally possible to have "neither" as a possibility. For example, you could have 0, 1, and 2 in the left-hand column of the chart instead of 1, 2, and 3. That would be solvable using the exact same method as in the video. But it won't work to have FOUR rows in the chart. For example, if we removed the phrase "every traveler has visited at least one" from Q3, the question would no longer be solvable. If you're bored and want to give it a try, go for it. :) Beyond that, I don't think there's any value in inventing tons of hypothetical versions of this question type. The GMAT can certainly be creative in coming up with new variations on questions, but these "three-way" overlapping sets questions are rare to begin with -- fewer than a dozen have ever been released publicly, out several thousand official quant questions. So it's somewhat unlikely that you'll see one of these on your exam at all -- and it's incredibly unlikely that you'll see a variation that goes beyond what we've described in the video. It's theoretically possible, but I wouldn't lose any sleep over it. I hope that helps a bit!

As in most GMAT Quant questions, there are probably multiple solution paths you could take to solve this problem. The method shown in the video is the way we recommend you solve this problem, but we'd be delighted to hear of others if you have your own way of getting to the right answer. I can think of one way of reasoning through this problem using probability formulas and the understanding that the probability of event X or Y occurring must be less than or equal to one, but that method doesn't provide the certainty that the method in the video provides so I'm not going to list it here. I'm also not going to work through that solution path as it involves probability formulas and this is the overlapping sets video. Let us know if you have an alternative method to solve this question, as we always love hearing of different ways to answer these questions!

Yup, you'll definitely hit a dead-end if you try to force Q3 into the type of double matrix that we used on Q1 and Q2. The double matrix only works if you have a pair of things that are both complementary -- meaning that 100% fit in exactly one of two categories. For example, in Q1, 100% of the population either eats varenyky or they eat no varenyky -- and logically, there are no exceptions. You fit one category or the other, but not both. So those are our two columns. Similarly, everybody either drinks Nemeroff, or they don't. So those are our rows. If you try to apply that same framework to Q3, it simply doesn't work. First of all, there are three countries, not two -- so there's no way to fit the data into the same little matrix. More importantly, those three countries are NOT complementary. We can't say that 100% of the population visits either Latvia OR Lithuania OR Estonia. It's possible to visit none, or exactly one, or all three. So the country names simply don't work as your columns -- and they won't work in a 2x2 matrix. That's why we use a completely different approach to Q3, as shown in the video. Bottom line: for the double matrix to work, your two columns need to consist of two complementary characteristics that account for 100% of the population, with no overlap. (The same is true of the rows.) If the data doesn't fit that criteria, the double matrix is unlikely to help much. I hope that helps a bit!

Thank you for the question, Grace! Charles here. I'll be brutally honest: that final explanation isn't the clearest thing I've ever put into a video, unfortunately. To be fair, that last question really doesn't fit terribly well into any of the standard methods that we might apply to overlapping sets questions (either of the chart-type things presented in this video, or Venn diagrams). But still, I should have been clearer about why I was using those bars. Here's how I would think about it: imagine that I have a grand total of 247 menu items. Now, I'm going to start "filling" the menus of 120 locations with those items. First, I'll give each location 1 new item. So I "filled" 120 menus with 1 item each -- and that's what the first bar represents. Then, I'll give each location a 2nd new item. So then I'm "filling" 120 menus with a 2nd item -- and that's what the next bar represents. At that point, I only have 7 new items left to distribute to the restaurants' menus. So only 7 locations need to have all 3 of the new items. And that's what the third and final bar represents. Let us know if that helps? Sorry again for the lack of clarity in the video! And don't worry about this one too much -- it's a rare and challenging variant that you're very, very unlikely to see on your actual exam.

@@GMATNinjaTutoring Charles, thanks for your explanations! I actually drew many Venn diagrams for this question but it is a pity that I cannot paste them here--you might find them amusing. I like using Venn diagrams because they are visual. My idea was that if the number of the locations that sell X or Y or Z is 120, 120 should be the result of 247 (X+Y+Z) minus the number of the locations that sell exactly two items, and minus twice the number of the locations that sell all three items. Since I wanted the latter to be as small as possible (I used the variable B to represent the number of locations that sell three items), the former needed to be as large as possible--and I thought that it should be 90-B. So I got one formula: 120=247-(90-B)-2B, and then I got B=37, which is quite different from your answer 7. I think I wrongly set 90-B, since your answer suggests that the number should be 120-B. Your approach is very straightforward, but could I check the following points? 1. When you say you give each of the 120 locations one new item, since no item is introduced by all 120 locations at the same time, could I understand that the first 120 menus should contain at least two items? (For example, the first 90 menus are X item, and the other 30 menus are Y item) 2. After you give each location the second batch of menu, since after this distribution, no location has three menus, the value of b is still 0. And I can imagine that the second batch of 120 menus as 55 menus of Y item and 65 menus of Z item. 3. Now, all locations have gotten two menus individually, but there are only 240 menus in total. The final seven menus (menus of Z item) have to be given to seven locations. So, seven stores would luckily get three types of menus and the value of b is 7. 4. I realize under this approach, if we order the stores from 1st to 120th and give menus in this order, not only the value of b is fixed at 7, the whole distribution situation would remain the same, regardless of which type of menus to be given first. If we give these stores menus of X items first, then menus of Y items and Z items, eventually 7 stores would sell all three items, 48 stores would sell exactly X items and Y items, 35 stores would sell X items and Z items, and 30 stores would sell Y items and Z items. But this distribution situation would not change even if we distribute the menus of Z items first, menus of X items and Y items later. I was very surprised to find out this. Thank you so much for providing this question and the explanations! I do think this question is rare, and so is the way of solution. Right now I cannot imagine how to apply this method to a typical overlapping-set problem. But it is fascinating and eye-opening! Thank you so much!

@@rejoicingGrace Haha, I'm glad that you're actually enjoying this, despite its lack of actual value on the GMAT! And yes, the logic you've laid out in #1-4 is correct. You don't really have to worry about the distinction between items X, Y, and Z here, interestingly.

@@GMATNinjaTutoring Ha it might be a bit painful but I am really impressed by these creative questions! Look forward to this Friday's session!

@@GMATNinjaTutoring Thank you for your explanation here. I too didn't understand the video solution but this text solution is lucid.

We certainly could, but I worry that it might be pretty redundant with this video. Three-way overlapping sets questions aren't terribly common on the GMAT to begin with, and when you do see them, they'll usually look like Q3 in this video. It's vaguely possible -- but pretty unlikely -- that you'll see a more exotic three-way overlapping sets question that looks a bit like #7 (which is arguably a two-way question in disguise) or #8. So if we did another video exclusively on the three-way questions, it would basically be repeating things that look a whole lot like #3, #7, and #8. I'm not sure that it would be very valuable for students, but if enough test-takers think it would be useful, we could give it a shot. :) Have fun studying, Ankush!

From the question, we know that 72 of the restaurants introduce item Z. This is the upper limit for the number of restaurants that introduce all three items as a restaurant can't introduce all three items without introducing item Z. This upper limit can be reached if all 72 of the restaurants that introduce item Z also introduce items X and Y. I hope that helps!

Did you get the answer?

There is double counting happening since total no. Of people not placing beach @1st rank is 54. Now when you look at the number of people placing either ski or city ahead, it comes to 66 people. This means there are some people who place ski AND city ahead of the beach. This means that 12 people ( 66 - 55) have to be common among the group of people placing ski and the group of people placing city before beach. Hope this helps

Thank you for the kind words! Thankfully, the situation you describe wouldn't happen. In a GMAT data sufficiency question, statement (1) and statement (2) might provide different information, but the scenario will remain consistent throughout the question. In this question, the box for 'approve of policy A' and 'neutral or disapprove of policy B' will be 175 throughout the entire question. You might not be able to find that value from the information provided, but you won't be given any information that changes the value. There is no chance the GMAT would make the value 175 with the information in statement (1) and 180 with the information in statement (2). I hope that helps!

Interesting solution path! It's not quite giving you the right answer though, and I think that's because you're omitting the possibility that a location introduces ZERO items. The question doesn't rule out that possibility. So it's very possible for 72 locations to introduce all 3 items. In that case, we can't possibly know exactly how many locations introduced 0, 1, or 2 items -- but we also don't really care, since the question is only concerned with the minimum and maximum number of locations that offer all 3 items. :) I hope that helps a bit!

Yes, it would!

Your Venn diagram method is a great alternative way of thinking about this question, thank you for sharing it! The two events cannot be disjoint given the answer you found for the minimum overlap. If we assume they were disjoint and the individual probabilities remained as they're written in the question then the total probability in the sample space would be at least 1.55. This is the minimum possible total probability in this scenario because we still have to add on the probability of neither event occurring. Since we can't have a total probability greater than 1, these two events cannot be disjoint. The maximum probability of both events occurring is 0.7, so the range of possible probabilities for both events occurring is 0.55 - 0.7. Since the product of the two probabilities (0.595) is within that range, the two events could be independent. However, we don't need to know whether they are independent or not to answer this question. I hope that helps!

If those 15 people have each visited 3 countries, then 15*3 = 45 countries have been visited. In this question, the final column represents the total number of countries that have been visited. If general if you're using this process, the final column represents the product of the values in the first and second columns. I hope that helps!

@@GMATNinjaTutoring Exactly the word "45 countries" is making me more confused. What I'm thinking is first of all only 3 countries are there how come 45 countries have been visited then? Next ,I think that those 3 countries have been visited 45 times. Please correct my thinking!

It might help to think of the 45 as representing the number of visits that have occurred. If 15 people have each visited 3 countries, then 45 visits have taken place. Does that make things more clear?

We can only find the combined probability of two events occurring by multiplying the individual probabilities if we know for certain that the two events are independent. If that was the case, the question would say something to tell you the events were independent. You wouldn't have to infer this; the question would explicitly tell you they were independent. I hope that helps!

The key on Q4 is the language in the question itself: all of the 200 residents eat jollof or suya or both. The important part: the phrasing suggests that everybody eats at least one of the foods -- so NONE of them eat neither. That's why I put a big zero in the center square of the chart. Try that, and see if it changes your answer? :) I hope that helps!

If 85 is the number of locations that introduce item Y, and 72 is the maximum number of locations that introduce all three items, then 85-72 tells us the number of locations that introduced item Y but didn't introduce all three items. I'm not sure how that's helpful or what it really tells us. In the text of this question, there is no indication that every location introduces at least one item. It's possible there are several locations that do not introduce an item at all. So, we could have 72 locations that introduce all three items, 13 that introduce items X and Y only, 5 that introduce item X only, and 30 that introduce none of the three items. This adds up to 120 and gives us the maximum number of locations that introduce all three items. I hope that helps!

Interesting solution path! I think it works fine in this situation, and if it works better for you than the solution in the video, that's great. I would argue that using the formula is far less intuitive for most test-takers, and the logic shown in the video is probably easier and more versatile. But this totally works, and when you're actually taking the exam, that's all you need!

@@GMATNinjaTutoring yes of course. the method in the video is more intuitive. I was just trying to connect the dots of my earlier learnings. cheers :)

Unfortunately, there's a small mistake in your arithmetic. If we assign X to 90 restaurants, then 30 do not have X (NX). We can then assign Y in a way that minimizes the chance of a restaurant having all three. This will give 30 (NX, Y), 55 (X, Y), and *35* (X, NY). This is the point where you make the mistake in your arithmetic as 30 + 55 + 45 = 130, which is too many restaurants. Assigning Z in a way that minimizes the number of restaurants that have all three items, we have 30 (NX, Y, Z) 35 (X, NY, Z) 48 (X, Y, NZ), and 7 (X, Y, Z). So the smallest number of restaurants that have all three items is 7. I hope that helps!

thanks so much for the prompt response! sorry for the confusion with my arithmetic error. I'll try watching again to try to understand your much more elegant solution intuitively @@GMATNinjaTutoring

My answer was like min of overcount of (3) = total counting - max overcount of (2) = 247- max overcount of (xy or xz or yz) = 247 - (85+72+72)=18. 18/3=6=b. So wrong.

The 45 in the solution wasn't chosen randomly. If 15 people each visit three countries, then 15*3 = 45 countries will be visited. In a similar fashion, Charles put an X in the table for the number of people that visited two countries each. If X people each visit two countries, then X*2 = 2X countries will be visited. And if (84 - X) people each visit one country, then (84 - X) * 1 = 84 - X countries will be visited. The numbers in this final column should sum to 123, as we know that's the number of "countings" or total number of countries that have been visited. I hope that helps!

@@GMATNinjaTutoringthankyou. 😳

Great question! First, what you're describing is highly common and totally normal. Nearly every GMAT student finds anxiety interfering with their test-taking process at some point. While there's no silver bullet solution, what you describe is definitely the right track -- take a step back, reset your brain, and try diving back in with renewed focus. Some test takers find this issue actually gets worse on test day, and can have a big impact on their score. If you suspect that might be the case for you, you may want to check out our article on tackling test anxiety: www.gmatninja.com/gmat/articles/basics/how-to-tackle-gmat-anxiety I hope that helps!

Unless we're watching you do the question, it's hard to know exactly why you missed the question. But as emphasized in the explanation in the video, it's very easy to misinterpret the question, especially if you're setting it up as a 3-way matrix. For most people, the key is recognizing that if a consumer "did not say that they approve of policy B", that's not the same thing as saying that they DISAPPROVE -- if they "did not say that they approve", they either disapproved or they're neutral. For more, check out the explanation starting at the 38:25 mark: kzfaq.info/get/bejne/hdl7fJBm35mvl2g.html. I hope that helps a bit!

Hi! Could you please explain what you mean when you say 'this is not working' for this question? Using the method Charles demonstrated in the video in question 3, I get an answer of 10 students registered for exactly two classes. If you get a different answer, could you please explain how you answered the question and I'll do what I can to help figure out where things are going wrong. Thank you!

The mistake occurs when you think that every one of the 120 locations must introduce one of the new items. Take a look at the bar diagram that Charles finishes drawing on the board at 55:15 -- in this scenario, 30 of the 120 locations introduce none of the three items, and this number will change depending on how you arrange which locations introduce items X, Y, and Z. This means you'll need another variable, let's say w, for the locations that introduce 0 items. However, I think the algebra is going to get very messy. I think you'll end up with something like y + 2z - w = 127 if you try to go down the same path, and maximizing z from that equation is going to get really difficult. I'm sorry your method didn't work out as it would have made a brilliant, neat solution. I hope that helps!

@@GMATNinjaTutoring but if we make case for 0 location as well how do we know this is a double counting problem then

I'm sorry, I'm not sure what you mean in your question -- what do you mean by a double counting problem? Could you please rephrase the question, and I'll do what I can to help? Thank you!

He's kinda funny-looking, too. - Every other GMAT Ninja tutor

Mine never stops spinning. 🤷🏻‍♂

If I do not approve of A, it doesn't necessarily mean that I disapprove of it. It might mean that I'm neutral about A. Similarly, if I do not approve of B, I might disapprove of B but I might just be neutral about it. So if I approve of neither A nor B, then I might disapprove of both of them, I could be neutral about both of them, or I could be neutral about one and disapprove of the other. The important thing is that I do not approve of A and I do not approve of B. Following that thought proess means we can fill in the "dis/neutral A" and "dis/neutral B" box with the 325 we're given in statement (2) and from there, we can fill in the rest of the table. This means statement (2) is sufficient to answer this question. I hope that helps!

@@GMATNinjaTutoring Thank you!

If someone did not say they approve of A, then they either disapprove of A or are neutral about it. Those who do not approve of A are already encapsulated in the 275 people we were given in the question, so there's no need for the +K. I hope that helps!

ah okay, thank you very much! I thought that if you don't say it means that you're neutral. @@GMATNinjaTutoring

Yup, it's a rough one! Don't lose any sleep over it at all -- as we get to the end of this particular videos, the questions get pretty nasty. They're a nice little workout if you're shooting for an elite quant score, but this is about as tough as things get on the GMAT. So there's no shame at all in struggling with those last few questions. Have fun studying, and thank you for watching!