Join us for MBA Spotlight – The Top 20 MBA Fair      Schedule of Events | Register

It is currently 04 Jun 2020, 13:59

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

A researcher plans to identify each participant in a certain medical

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Director
Director
User avatar
P
Status: Professional GMAT Tutor
Affiliations: AB, cum laude, Harvard University (Class of '02)
Joined: 10 Jul 2015
Posts: 776
Location: United States (CA)
Age: 40
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 medical  [#permalink]

Show Tags

New post 25 May 2016, 17:47
Attached is a visual that should help.
Attachments

Screen Shot 2016-05-25 at 6.44.12 PM.png
Screen Shot 2016-05-25 at 6.44.12 PM.png [ 64.3 KiB | Viewed 1825 times ]

Target Test Prep Representative
User avatar
V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 10654
Location: United States (CA)
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 15 Jun 2016, 05:06
3
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

Scott@TargetTestPrep.com
TTP - Target Test Prep Logo
202 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Intern
Intern
avatar
B
Joined: 09 Mar 2017
Posts: 32
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
User avatar
V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
User avatar
S
Status: It's now or never
Joined: 10 Feb 2017
Posts: 169
Location: India
GMAT 1: 650 Q40 V39
GPA: 3
WE: Consulting (Consulting)
GMAT ToolKit User
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
User avatar
V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 03 Oct 2017, 08:48
Director
Director
User avatar
P
Status: Professional GMAT Tutor
Affiliations: AB, cum laude, Harvard University (Class of '02)
Joined: 10 Jul 2015
Posts: 776
Location: United States (CA)
Age: 40
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 medical  [#permalink]

Show Tags

New post 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

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.
GMAT Club Legend
GMAT Club Legend
User avatar
V
Joined: 11 Sep 2015
Posts: 4887
Location: Canada
GMAT 1: 770 Q49 V46
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
Image
GMAT Club Legend
GMAT Club Legend
User avatar
V
Joined: 11 Sep 2015
Posts: 4887
Location: Canada
GMAT 1: 770 Q49 V46
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
Image
VP
VP
User avatar
D
Joined: 09 Mar 2016
Posts: 1241
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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 I`ve 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
User avatar
V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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 I`ve 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.
_________________
Intern
Intern
avatar
B
Joined: 11 Sep 2017
Posts: 33
Schools: IMD '21
GMAT 1: 740 Q50 V40
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
User avatar
V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16775
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 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
_________________
Contact Rich at: Rich.C@empowergmat.com
Image


The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Intern
Intern
avatar
B
Joined: 15 Feb 2018
Posts: 3
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 06 Nov 2018, 09:07
Bunuel
A,B,AD,C,BC,ABC,D,CD,ABCD,AD,AC,BD
so effectively we have used 4 letters.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 64275
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 06 Nov 2018, 09:11
1
Senior Manager
Senior Manager
User avatar
P
Joined: 05 Feb 2018
Posts: 440
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 28 Jan 2019, 07:50
1
Got tripped up a bit on the wording, but saw the familiarity of the problem. The hard part is that these combination/permutation word problems express things in an particular way that doesn't always make it immediately clear whether order matters or not.

The issue I had was with the use of the word order, which in my mind would typically be associated with permutations, but the restriction of the problem actually means it's a combination.

"written in alphabetical order" - meaning that the letters have to be alphabetically ordered, but not necessarily consecutively next to each other on the alphabet. This actually makes it a combination and not a permutation question, because it removes all the other possibilities (i.e. we can have AB but not BA). Drawing it out makes sense:

Image

The fact that the letters have to be ordered alphabetically is a convoluted way of saying only unique combinations of letters matter. Hopefully my thinking is correct in this regard.
Manager
Manager
User avatar
S
Joined: 21 Jul 2018
Posts: 173
Reviews Badge
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 02 Apr 2019, 07:13
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


Bunuel, chetan2u

Can any one elaborate condition of highlighted part.
GMAT Club Legend
GMAT Club Legend
User avatar
V
Joined: 11 Sep 2015
Posts: 4887
Location: Canada
GMAT 1: 770 Q49 V46
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 02 Apr 2019, 08:05
2
Top Contributor
Gmatprep550 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


Bunuel, chetan2u

Can any one elaborate condition of highlighted part.


It means that the letters we choose must be presented in alphabetical order.
So, for example, EG is good, since E comes before G in the alphabet.
Conversely, GE is not acceptable, since G does not come before E in the alphabet.

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Image
Intern
Intern
avatar
Joined: 04 Feb 2014
Posts: 1
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 29 May 2019, 00:02
Bunuel, can you explain why this is a combination and not a permutation?


Bunuel wrote:
Almost identical question:

John has 12 clients and he wants to use color coding to identify each client. If either a single color or a pair of two different colors can represent a client code, what is the minimum number of colors needed for the coding? Assume that changing the color order within a pair does not produce different codes.
A. 24
B. 12
C. 7
D. 6
E. 5

The concept is not that hard. We can use combination or trial and error approach.

Combination approach:
Let # of colors needed be \(n\), then it must be true that \(n+C^2_n\geq{12}\) (\(C^2_n\) - # of ways to choose the pair of different colors from \(n\) colors when order doesn't matter) --> \(n+\frac{n(n-1)}{2}\geq{12}\) --> \(2n+n(n-1)\geq{24}\) --> \(n(n+1)\geq{24}\) --> as \(n\) is an integer (it represents # of colors) \(n\geq{5}\) --> \(n_{min}=5\).

Trial and error approach:
If the minimum number of colors needed is 4 then there are 4 single color codes possible PLUS \(C^2_4=6\) two-color codes --> 4+6=10<12 --> not enough for 12 codes;

If the minimum number of colors needed is 5 then there are 5 single color codes possible PLUS \(C^2_5=10\) two-color codes --> 5+10=15>12 --> more than enough for 12 codes.

Actually as the least answer choice is 5 then if you tried it first you'd get the correct answer right away.

Answer: E.

Hope it helps.
EMPOWERgmat Instructor
User avatar
V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16775
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: A researcher plans to identify each participant in a certain medical  [#permalink]

Show Tags

New post 29 May 2019, 18:32
Hi crushorange,

The last sentence in the prompt tells us: "Assume that changing the color order within a pair does NOT produce different codes." This means that two different colors will only produce ONE code. For example Blue-Green and Green-Blue are the SAME code. Thus, the order of the colors does NOT matter and we're dealing with a Combination.

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com
Image


The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
GMAT Club Bot
Re: A researcher plans to identify each participant in a certain medical   [#permalink] 29 May 2019, 18:32

Go to page   Previous    1   2   3   4    Next  [ 66 posts ] 

A researcher plans to identify each participant in a certain medical

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne