GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 19 Aug 2019, 15:43

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 registration number is made using 3 letters of the English alphabets

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

Hide Tags

Find Similar Topics 
Senior Manager
Senior Manager
User avatar
G
Joined: 16 Jan 2019
Posts: 409
Location: India
Concentration: General Management
WE: Sales (Other)
A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post Updated on: 22 Jun 2019, 14:14
7
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

26% (02:58) correct 74% (03:11) wrong based on 19 sessions

HideShow timer Statistics

A registration number is made using 3 letters of the English alphabet followed by three numerical digits. The letters can be in any order but the numbers should be in a non-increasing order - the second digit is not greater than the first, and the third digit is not greater than the second digit. How many such registration numbers can be formed?

A. \(26^3*240\)

B. \(26^3*220\)

C. \(26C3*240\)

D. \(26C3*220\)

E. \(26^3*320\)

Originally posted by firas92 on 21 Jun 2019, 23:13.
Last edited by firas92 on 22 Jun 2019, 14:14, edited 1 time in total.
Senior Manager
Senior Manager
avatar
P
Joined: 05 Jul 2018
Posts: 346
Location: India
Concentration: General Management, Technology
GMAT 1: 600 Q47 V26
GRE 1: Q162 V149
GPA: 3.6
WE: Information Technology (Consulting)
Reviews Badge CAT Tests
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 00:11
Struggling with the statement: numbers should be in non-increasing order. Can someone help me find such 3 digit numbers . The other part should obviously be 26^3
_________________
GMAT Tutor
avatar
G
Joined: 24 Jun 2008
Posts: 1816
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 08:55
We have 26 choices for each letter, so 26^3 choices for the first three letters.

For the three non-increasing digits, there are three possibilities:

- the three digits are identical: then we have 10 possibilities (000, 111, 222, etc)
- two of the digits are identical and one different. We then have 10 ways to choose the identical pair, and 9 remaining ways to choose the other digit, for 90 possible choices. Once we choose the three digits, there will only be one order we can put them in that will be non-increasing, so we have 90 possibilities in this case.
- all three digits are different. Then we have (10)(9)(8)/3! = 120 ways to choose the three digits, and once we've chosen the digits, only one order we can put them in will be non-increasing. So we have 120 possibilities in this case.

So there are 10+90+120 = 220 ways to make the number, and 26^3 * 220 registration numbers in total.

There's a much faster way to choose how to count the three digits, but the method uses partitions in a complicated way, and so is well beyond GMAT level math.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
GMAT Club Legend
GMAT Club Legend
User avatar
D
Joined: 18 Aug 2017
Posts: 4476
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member CAT Tests
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 09:18
IanStewart could you please explain the highlighted part of your solution ; my query is while choosing a three digit no where 2 of the digits are identical so in that case shouldnt the combination be 10*1*9* 3 ; for eg ; 122, 212,221 we can get three different nos.
the question though has stated that "the numbers should be in a non-increasing order"
eg ; 571 , 481 etc. how to identify such no? is the approach of 10*1*9 ; 90 correct ?

IanStewart wrote:
We have 26 choices for each letter, so 26^3 choices for the first three letters.

For the three non-increasing digits, there are three possibilities:

- the three digits are identical: then we have 10 possibilities (000, 111, 222, etc)
- ttwo of the digits are identical and one different. We then have 10 ways to choose the identical pair, and 9 remaining ways to choose the other digit, for 90 possible choices. Once we choose the three digits, there will only be one order we can put them in that will be non-increasing, so we have 90 possibilities in this case.
- all three digits are different. Then we have (10)(9)(8)/3! = 120 ways to choose the three digits, and once we've chosen the digits, only one order we can put them in will be non-increasing. So we have 120 possibilities in this case.

So there are 10+90+120 = 220 ways to make the number, and 26^3 * 220 registration numbers in total.

There's a much faster way to choose how to count the three digits, but the method uses partitions in a complicated way, and so is well beyond GMAT level math.

_________________
If you liked my solution then please give Kudos. Kudos encourage active discussions.
GMAT Tutor
avatar
G
Joined: 24 Jun 2008
Posts: 1816
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 09:42
Archit3110 wrote:
my query is while choosing a three digit no where 2 of the digits are identical so in that case shouldnt the combination be 10*1*9* 3 ; for eg ; 122, 212,221 we can get three different nos.
the question though has stated that "the numbers should be in a non-increasing order"
eg ; 571 , 481 etc. how to identify such no? is the approach of 10*1*9 ; 90 correct ?


I see -- the wording of the question is ambiguous. In my solution, when they say "non-increasing order", I've interpreted that to mean "the digits never increase". So in my solution, 841 is in non-increasing order, but 481 is not -- the digits increase from 4 to 8, then decrease from 8 to 1.

I think you're interpreting "non-increasing order" to mean "not in strictly increasing order". Under your interpretation, 481 is "non-increasing", and so is 841, but 148 is not. If you interpret the question in that way, the answer will be much larger.

The wording of the question simply isn't clear, so both of our interpretations make logical sense - there's no way to tell even what the question is asking us for, so no way to say what the "right" answer is. But under my interpretation, once you choose a pair of digits and a third different digit, which you can do in 90 ways, there is then only one never-increasing order you can put the digits in. If you choose 2, 2, and 1, the only number you can make with never-increasing digits is 221.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
GMAT Club Legend
GMAT Club Legend
User avatar
D
Joined: 18 Aug 2017
Posts: 4476
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member CAT Tests
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 11:03
1
yes you are correct IanStewart question wording is kinda ambiguous

Hey firas92 could you please share the source of this question ?

IanStewart wrote:
Archit3110 wrote:
my query is while choosing a three digit no where 2 of the digits are identical so in that case shouldnt the combination be 10*1*9* 3 ; for eg ; 122, 212,221 we can get three different nos.
the question though has stated that "the numbers should be in a non-increasing order"
eg ; 571 , 481 etc. how to identify such no? is the approach of 10*1*9 ; 90 correct ?


I see -- the wording of the question is ambiguous. In my solution, when they say "non-increasing order", I've interpreted that to mean "the digits never increase". So in my solution, 841 is in non-increasing order, but 481 is not -- the digits increase from 4 to 8, then decrease from 8 to 1.

I think you're interpreting "non-increasing order" to mean "not in strictly increasing order". Under your interpretation, 481 is "non-increasing", and so is 841, but 148 is not. If you interpret the question in that way, the answer will be much larger.

The wording of the question simply isn't clear, so both of our interpretations make logical sense - there's no way to tell even what the question is asking us for, so no way to say what the "right" answer is. But under my interpretation, once you choose a pair of digits and a third different digit, which you can do in 90 ways, there is then only one never-increasing order you can put the digits in. If you choose 2, 2, and 1, the only number you can make with never-increasing digits is 221.

_________________
If you liked my solution then please give Kudos. Kudos encourage active discussions.
Senior Manager
Senior Manager
User avatar
G
Joined: 16 Jan 2019
Posts: 409
Location: India
Concentration: General Management
WE: Sales (Other)
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 11:15
Archit3110 wrote:
yes you are correct IanStewart question wording is kinda ambiguous

Hey firas92 could you please share the source of this question ?

IanStewart wrote:
Archit3110 wrote:
my query is while choosing a three digit no where 2 of the digits are identical so in that case shouldnt the combination be 10*1*9* 3 ; for eg ; 122, 212,221 we can get three different nos.
the question though has stated that "the numbers should be in a non-increasing order"
eg ; 571 , 481 etc. how to identify such no? is the approach of 10*1*9 ; 90 correct ?


I see -- the wording of the question is ambiguous. In my solution, when they say "non-increasing order", I've interpreted that to mean "the digits never increase". So in my solution, 841 is in non-increasing order, but 481 is not -- the digits increase from 4 to 8, then decrease from 8 to 1.

I think you're interpreting "non-increasing order" to mean "not in strictly increasing order". Under your interpretation, 481 is "non-increasing", and so is 841, but 148 is not. If you interpret the question in that way, the answer will be much larger.

The wording of the question simply isn't clear, so both of our interpretations make logical sense - there's no way to tell even what the question is asking us for, so no way to say what the "right" answer is. But under my interpretation, once you choose a pair of digits and a third different digit, which you can do in 90 ways, there is then only one never-increasing order you can put the digits in. If you choose 2, 2, and 1, the only number you can make with never-increasing digits is 221.


Archit3110

https://cracku.in/

I guess this is not typically a source for GMAT questions.
Senior Manager
Senior Manager
User avatar
G
Joined: 16 Jan 2019
Posts: 409
Location: India
Concentration: General Management
WE: Sales (Other)
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 22 Jun 2019, 11:21
1
OFFICIAL EXPLANATION

The letters can be picked in 26*26*26 ways

A non increasing set of numbers means that a digit that follows another digit cannot be more in value than the preceding digit.

In order to select the numbers in a non increasing sequence,

If all three are distinct, the number of cases is 10C3=120

If exactly two of the three digits are the same, the number of ways of selecting the two digits is 10C2 each of which can be arranged in 2 ways. (Numbers like 221, 211 fall in this category). So this total is 10C2*2=90

The last case is if all three numbers were the same. There are 10 ways (000~999)

So the total number of ways of picking the digits is 120+90+10=220

Hence the number of possible registration numbers is \(26^3*220\)
Senior Manager
Senior Manager
avatar
P
Joined: 05 Jul 2018
Posts: 346
Location: India
Concentration: General Management, Technology
GMAT 1: 600 Q47 V26
GRE 1: Q162 V149
GPA: 3.6
WE: Information Technology (Consulting)
Reviews Badge CAT Tests
Re: A registration number is made using 3 letters of the English alphabets  [#permalink]

Show Tags

New post 23 Jun 2019, 01:41
First 3 characters is a no brainer... 26^3 ways to do that

Last 3 char is tricky

Non increasing 3 digit codes

Possibility 1: all repeating digits =10

Possibility 2 : 2 digits repeating = 10*9 (10 for repeating and excluding the treated digits 1 digit remain for the other 3rd option)=90

Possibility 3: No repeating digits = (10*9*8)3! =120

Why divide by 3 factorial ..because there is only one way in which each paid of 3 digits can be (the non increasing way)

Total no of.codes = 26^3 *(10+90+120)

=26^3 * 220

Posted from my mobile device
_________________
GMAT Club Bot
Re: A registration number is made using 3 letters of the English alphabets   [#permalink] 23 Jun 2019, 01:41
Display posts from previous: Sort by

A registration number is made using 3 letters of the English alphabets

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





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