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# A researcher plans to identify each participant in a certain

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Re: A researcher plans to identify each participant in a certain  [#permalink]

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21 Nov 2015, 23:13
Hi mvictor,

There's a difference between alphabetical order and CONSECUTIVE alphabetical order (in the same way that there's a difference between putting integers in numerical order and dealing with consecutive integers).

As an example, when dealing with the letters A, B, C and D there are 6 different pairs of letters that you could put in alphabetical order:

AB
AC
BC
BD
CD

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# Rich Cohen

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** Director Status: Professional GMAT Tutor Affiliations: AB, cum laude, Harvard University (Class of '02) Joined: 10 Jul 2015 Posts: 672 Location: United States (CA) Age: 39 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GRE 1: Q168 V169 WE: Education (Education) Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 25 May 2016, 17:47 Attached is a visual that should help. Attachments Screen Shot 2016-05-25 at 6.44.12 PM.png [ 64.3 KiB | Viewed 1161 times ] _________________ Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching worldwide since 2002. One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong). You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y94hlarr Date of Birth: 09 December 1979. GMAT Action Plan and Free E-Book - McElroy Tutoring Contact: mcelroy@post.harvard.edu (I do not respond to PMs on GMAT Club.) ...or find me on Reddit: http://www.reddit.com/r/GMATpreparation Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 4317 Location: United States (CA) Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 15 Jun 2016, 05:06 1 sarb wrote: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8 Let's use the answer choices to help us solve this problem. We are looking for the minimum number of letters that can be used. The smallest number from the answer choices is 4, so let’s ask ourselves this question: Can we use only 4 letters to represent the 12 participants? Assume that the 4 letters are A, B, C and D (keep in mind that for each participant we can use either one letter OR two letters to represent him or her; however if we use two letters, they must be in alphabetical order): 1) A 2) B 3) C 4) D 5) AB 6) AC 7) AD 8) BC 9) BD 10) CD Under this scheme, we can represent only 10 of the 12 participants. So let's add in one more letter, say E, and see if having an additional letter allows us to have a unique identifier for each of the 12 participants: 1) A 2) B 3) C 4) D 5) AB 6) AC 7) AD 8) BC 9) BD 10) CD 11) E 12) AE As you can see, once we’ve added in the letter E we can represent all 12 participants. Since we’ve used A, B, C, D and E, the minimum number of letters that can be used is 5. Answer B _________________ Scott Woodbury-Stewart Founder and CEO GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Current Student Joined: 31 Jan 2016 Posts: 21 Schools: Rotman '19 (A) A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 17 Aug 2016, 07:45 Hi, Could someone please explain how we went from n!/2!(n-2)! -> n(n-1)/2 ? In addition to the above query, I am still unclear as to why we use combination formula to solve this particular question when order clearly matters? I read Bunuels response as to why, but I am still unclear. I must have spent 5 days reading the explanation. Posted from my mobile device Manager Joined: 07 Jul 2016 Posts: 78 GPA: 4 A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 17 Aug 2016, 09:36 1 g3lo18 wrote: In addition to the above query, I am still unclear as to why we use combination formula to solve this particular question when order clearly matters? I read Bunuels response as to why, but I am still unclear. I must have spent 5 days reading the explanation. Posted from my mobile device (replying here instead of in chat as there's an MBA session going on). The only thing that matters is the number of distinct pairs. Stating that the order must be alphabetical states that only 1 of all potential pairs is valid. This is equivalent to the combination formula as shown below. For example, with $$\{A,B,C,D\}$$, the list of permutations of size 2 is $$4_P 2 = 12$$ AB, BA = 2! ways to arrange 2 elements AC, CA AD, DA BC, CB BD, DB CD, DC Saying that the order of the selection does not matter is equivalent to saying that every permutation is the same which is equivalent to saying that only one of each permutation is valid. Permutations = $$\frac{n!}{(n-k)!}$$ Combinations = $$\frac{n!}{k!(n-k)!}$$ Number of ways of arranging each new selection of elements = $$k!$$ Where 1 out of every new selection is valid = $$\frac{1}{k!}$$ Permutations where only 1 permutation is valid = $$\frac{n!}{(n-k)!} \times \frac{1}{k!} =$$ Combinations _________________ Please press +1 Kudos if this post helps. Current Student Joined: 31 Jan 2016 Posts: 21 Schools: Rotman '19 (A) Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 17 Aug 2016, 09:53 Thank you for that. I guess my confusion stems from watching the GMATPrepNow videos. Brent explains that if the order does not matter, we use nCr formula. However, in this case it does matter (alphabetical). I guess there may be an error in his videos? Manager Joined: 07 Jul 2016 Posts: 78 GPA: 4 A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 17 Aug 2016, 10:22 g3lo18 wrote: Thank you for that. I guess my confusion stems from watching the GMATPrepNow videos. Brent explains that if the order does not matter, we use nCr formula. However, in this case it does matter (alphabetical). I guess there may be an error in his videos? Sorry, this is relatively hard to explain and I don't feel I'm making a good job of explaining it. In this question, the order does not matter. The problem lies with the interpretation of the phrase "the order matters". Specifically it means: when selecting a set of elements for the output, every permutation of that set is valid. This means that $$\{A,B,D\}$$ and $$\{D,A,B\}$$ are both distinct and valid elements in the solution set. A combination on the other hand specifies that $$\{A,B,D\}$$ and $$\{D,A,B\}$$ are equivalent (which can also be described by saying that the order does not matter). This means that for the set of possible solutions containing three elements, only one of which is valid. In this question, every pair of letters maps to a single solution: out of $$\{\{A,B\},\{B,A\}\}$$, only $$\{A,B\}$$ is valid. We are performing the operation of moving from a set of elements to a single element. Therefore we use combinations. EDIT (may be more clear): In this question, we are first taking a collection of letters and determining how many pairs of distinct letters we can make (permutations). For every pair of distinct letters, we are then mapping from that collection to a single element. This map from a collection to a single element is a combination (all elements in the set of $$\{\{A,B\},\{B,A\}\}$$ are not a valid solutions, but each is a map to the single valid solution in the set: $$\{A,B\})$$. There is a good video explaining the concept here (the example at 5:01): https://www.coursera.org/learn/fe-exam/ ... mbinations I hope this helps (and sorry that I can't explain things better). _________________ Please press +1 Kudos if this post helps. Intern Joined: 09 Mar 2017 Posts: 36 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 10 May 2017, 05:33 Bunuel wrote: sarb wrote: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8 Say there are minimum of $$n$$ letters needed, then; The # of single letter codes possible would be $$n$$ itself; The # of pair of distinct letters codes possible would be $$C^2_n$$ (in alphabetical order); We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$. Answer: B. Hope it's clear. Can someone explain what this $$C^2_n+n\geq{12}$$ means? I also saw a reference to the same type of symbol with an A instead, what does that mean? Math Expert Joined: 02 Sep 2009 Posts: 51280 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 10 May 2017, 05:37 brandon7 wrote: Bunuel wrote: sarb wrote: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8 Say there are minimum of $$n$$ letters needed, then; The # of single letter codes possible would be $$n$$ itself; The # of pair of distinct letters codes possible would be $$C^2_n$$ (in alphabetical order); We want $$C^2_n+n\geq{12}$$ --> $$\frac{n(n-1)}{2}+n\geq{12}$$ --> $$n(n-1)+2n\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> $$n_{min}=5$$. Answer: B. Hope it's clear. Can someone explain what this $$C^2_n+n\geq{12}$$ means? I also saw a reference to the same type of symbol with an A instead, what does that mean? C stands for combinations: $$C^2_n=\frac{n!}{2!(n-2)!}$$ Combinatorics Made Easy! Theory on Combinations DS questions on Combinations PS questions on Combinations Hope it helps. _________________ Mannheim Thread Master Status: It's now or never Joined: 10 Feb 2017 Posts: 189 Location: India GMAT 1: 650 Q40 V39 GPA: 3 WE: Consulting (Consulting) Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 03 Oct 2017, 08:28 Hi Bunuel, Can you please share the number of codes possible in alphabets. I am still confused with the combinations between 5 alphabets and assigning 12 unique codes with two distinct letters. Thanks. _________________ 2017-2018 MBA Deadlines Threadmaster for B-school Discussions Class of 2019: Mannheim Business School Class 0f 2020: HHL Leipzig Math Expert Joined: 02 Sep 2009 Posts: 51280 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 03 Oct 2017, 08:48 AnubhavK wrote: Hi Bunuel, Can you please share the number of codes possible in alphabets. I am still confused with the combinations between 5 alphabets and assigning 12 unique codes with two distinct letters. Thanks. The number of codes consisting of either a single letter or a pair of distinct letters from 26-letter alphabet is $$26+C^2_{26}$$. _________________ Director Status: Professional GMAT Tutor Affiliations: AB, cum laude, Harvard University (Class of '02) Joined: 10 Jul 2015 Posts: 672 Location: United States (CA) Age: 39 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GRE 1: Q168 V169 WE: Education (Education) A researcher plans to identify each participant in a certain [#permalink] ### Show Tags Updated on: 29 Nov 2017, 18:47 Top Contributor Since the question asks for the minimum number of letters, it makes sense to start with the least answer (choice A) and work your way down. A) 4 choose 1 is 4, and 4 choose 2 is 6. Unfortunately this only adds up to 10, and 10 < 12. B) 5 choose 1 is 5 and 5 choose 2 is 10. This adds up to 15, and 15 > 12. We have a winner! -Brian _________________ Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching worldwide since 2002. One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong). You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y94hlarr Date of Birth: 09 December 1979. GMAT Action Plan and Free E-Book - McElroy Tutoring Contact: mcelroy@post.harvard.edu (I do not respond to PMs on GMAT Club.) ...or find me on Reddit: http://www.reddit.com/r/GMATpreparation Originally posted by mcelroytutoring on 17 Oct 2017, 11:09. Last edited by mcelroytutoring on 29 Nov 2017, 18:47, edited 1 time in total. CEO Joined: 11 Sep 2015 Posts: 3243 Location: Canada A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 06 Dec 2017, 08:05 1 Top Contributor 1 sarb wrote: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8 One approach is to add a BLANK to the letters in order to account for the possibility of using just one letter for a code. ASIDE: Notice that, if we select 2 characters, there's only 1 possible code that can be created. The reason for this is that the 2 characters must be in ALPHABETICAL order. Or, in the case that a letter and a blank are selected, there's only one possible code as well. Now we'll test the answer choices. Answer choice A (4 letters) Let the letters be A, B, C, D We'll add a "-" to represent a BLANK. So, we must choose 2 characters from {A, B, C, D, -} In how many ways can we select 2 characters? We can use combinations to answer this. There are 5 characters, and we must select 2. This can be accomplished in 5C2 ways (= 10 ways). So, there are only 10 possible codes if we use 4 letters. We want at least 12 codes. [i]ASIDE: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head below. Answer choice B (5 letters) Let the letters be A, B, C, D, E Once again, we'll add a "-" to represent a BLANK. So, we must choose 2 characters from {A, B, C, D, E, -} There are 6 characters, and we must select 2. This can be accomplished in 6C2 ways (= 15 ways...PERFECT). So, the least number of characters needed is 5 Answer: B RELATED VIDEO _________________ Test confidently with gmatprepnow.com CEO Joined: 11 Sep 2015 Posts: 3243 Location: Canada Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 06 Dec 2017, 08:08 1 Top Contributor sarb wrote: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8 We can also TEST each answer choice by LISTING all possible codes. Answer choice A (4 letters) Let the letters be A, B, C, D The possible codes are: A B C D AB AC AD BC BD CD TOTAL = 10 (not enough. We need at least 12 codes) Answer choice B (5 letters) Let the letters be A, B, C, D, E The possible codes are: A B C D E AB AC AD AE BC BD BE CD CE DC TOTAL = 15 Perfect, 5 letters will give us the 12 codes we need. Answer: B Cheers, Brent _________________ Test confidently with gmatprepnow.com Intern Joined: 16 Jul 2016 Posts: 34 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 19 Dec 2017, 10:39 Let the first spot be A. We have to use another letter for the next slot so let it be B. For the next we have to ask ourselves can we make an entry without introducing another letter? we can. so far we have A, B, AB,____. For the fourth space must we introduce a letter? the answer is yes. so we have A, B, AB, C, ____,______ for the fifth and sixth space we can have AC and BC A, B, AB, C, AC, BC,____,______,______ We must introduce another letter so let it be D A, B, AB, C, AC, BC, D, AD, BD, CD, ____,_____ We have to introduce another letter so let it be E. A, B, AB, C, AC, BC, D, AD, BD, CD, E, AE, BE, CE, DE We can go up to fifteen. We need 12 so the total different letters we used were A, B, C, D, and E. The answer is 5. VP Joined: 09 Mar 2016 Posts: 1251 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 19 Dec 2017, 14:26 Bunuel wrote: kevn1115 wrote: Hi Bunuel, I'm confused on when you show that n! = (n-2)!*(n-1)*n...why is n! only limited to those 3 factors? I guess the question is why do you start at (n-2)!? Thanks.! $$C^2_n=\frac{n!}{2!(n-2)!}$$. Now, notice that $$n!=(n-2)!*(n-1)*n$$, hence $$C^2_n=\frac{n!}{2!(n-2)!}=\frac{(n-2)!*(n-1)*n}{2!(n-2)!}=\frac{(n-1)n}{2}$$. Hope it's clear. n! is the product of positive integers from 1 to n, inclusive: n! = 1*2*...*(n-4)*(n-3)(n-2)(n-1)n. To simplify $$\frac{n!}{2!(n-2)!}$$ I wrote n! as (n-2)!*(n-1)*n this enables us to reduce by (n-2)! to get $$\frac{(n-1)n}{2}$$. Hope it's clear. Bunuel -Thank you ! Now when Ive reviewed the whole thread and still trying to understand some moments - why do you write it in this order 1*2*...*(n-4)*(n-3)(n-2)(n-1)n and not vice versa like this 1*2*...n(n-1)(n-2)(n-3)(n-4) etc ...also why you say" notice that n!=(n-2)(n-1)n" yes it as n! is in numerator as per formula and unlike formula, you simplify n! = n!/2!(n-2)! into != (n-2)(n-1)n" / 2!(n-2)! first cant not "notice" the important detail you are trying to imply by pointing at this ---> n!=(n-2)(n-1)n - can I be helped with further explanation to understand this "notice" because in other combination formulas we didn't apply such simplification Math Expert Joined: 02 Sep 2009 Posts: 51280 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 19 Dec 2017, 19:36 1 dave13 wrote: Bunuel wrote: kevn1115 wrote: Hi Bunuel, I'm confused on when you show that n! = (n-2)!*(n-1)*n...why is n! only limited to those 3 factors? I guess the question is why do you start at (n-2)!? Thanks.! $$C^2_n=\frac{n!}{2!(n-2)!}$$. Now, notice that $$n!=(n-2)!*(n-1)*n$$, hence $$C^2_n=\frac{n!}{2!(n-2)!}=\frac{(n-2)!*(n-1)*n}{2!(n-2)!}=\frac{(n-1)n}{2}$$. Hope it's clear. n! is the product of positive integers from 1 to n, inclusive: n! = 1*2*...*(n-4)*(n-3)(n-2)(n-1)n. To simplify $$\frac{n!}{2!(n-2)!}$$ I wrote n! as (n-2)!*(n-1)*n this enables us to reduce by (n-2)! to get $$\frac{(n-1)n}{2}$$. Hope it's clear. Bunuel -Thank you ! Now when Ive reviewed the whole thread and still trying to understand some moments - why do you write it in this order 1*2*...*(n-4)*(n-3)(n-2)(n-1)n and not vice versa like this 1*2*...n(n-1)(n-2)(n-3)(n-4) etc ...also why you say" notice that n!=(n-2)(n-1)n" yes it as n! is in numerator as per formula and unlike formula, you simplify n! = n!/2!(n-2)! into != (n-2)(n-1)n" / 2!(n-2)! first cant not "notice" the important detail you are trying to imply by pointing at this ---> n!=(n-2)(n-1)n - can I be helped with further explanation to understand this "notice" because in other combination formulas we didn't apply such simplification 1. n! is the product of integers from 1 to n, inclusive. So, n! = 1*2*...*(n-4)(n-3)(n-2)(n-1)n (1 is the smallest and n is the largest). Yes, you can write this in any order but it does not change anything. 2. n! = (n-2)!*(n-1)*n because (n-2)! = 1*2*...*(n-4)(n-3)(n-2), so (n-2)!*(n-1)*n = [1*2*...*(n-4)(n-3)(n-2)](n-1)n = n! 3. We can write $$C^2_n=\frac{n!}{2!(n-2)!}=\frac{(n-2)!*(n-1)*n}{2!(n-2)!}=\frac{(n-1)n}{2}$$ whenever it's necessary. Hope it's clear now. _________________ Retired Moderator Joined: 26 Nov 2012 Posts: 591 A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 09 Jan 2018, 07:12 sarb wrote: A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8 We can use just logic to solve this question. Here we are given that all letters and we need to get the least number of letters for 12 participants and all the letter should in alphabetical order. Let's choose 4 letters since this is the least number. A B C D Then we can choose two letters again , AB, AC, AD, BC, BD, BD - total 6 ways. ( Note here we can't BA since we need to follow alphabetical order ) Total 4 + 6 = 10 and this suits only for 10 participants. Let's choose 5 letters. A B C D E - 5 ways AB AC AD AE BC BD BE CD CE DE - 10 ways . We got 15 ways and 15 different codes are sufficient to give the code for 12 participants. Hence 5. Intern Joined: 11 Sep 2017 Posts: 29 GMAT 1: 740 Q50 V40 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 02 Oct 2018, 02:31 The question says that the letters have to be in alphabetical order so in this case can we keep AC too. Since while writing AC we are missing B and not writing in alphabetical order EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13108 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: A researcher plans to identify each participant in a certain [#permalink] ### Show Tags 02 Oct 2018, 10:57 Hi hassu13, There's a difference between alphabetical order and CONSECUTIVE alphabetical order (in the same way that there's a difference between putting integers in numerical order and dealing with consecutive integers). As an example, when dealing with the letters A, B, C and D there are 6 different pairs of letters that you could put in alphabetical order: AB AC AD BC BD CD GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: A researcher plans to identify each participant in a certain &nbs [#permalink] 02 Oct 2018, 10:57

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