It is currently 04 Sep 2026, 10:42
Home
Messages
Tests
Schools
Events
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 04 Sep 2026
Posts: 113,146
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,146
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
04 Dec 2020, 05:15
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
Official Answer and Stats are available only to registered users. Register/Login.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

85% (hard)

Question Stats:

53% (02:47) correct 47% (02:42) wrong based on 201 sessions

History

Date
Time
Result
Not Attempted Yet
A book contains 732 pages numbered from 1 to 732. If a student randomly opens the book, what is the probability that the page number contains the digit 1?

A. 112/732

B. 184/732

C. 227/732

D. 233/732

E. 254/732

Are You Up For the Challenge: 700 Level Questions
ShowHide Answer
Official Answer
Official Answer and Stats are available only to registered users. Register/Login.

_________________
New to the GMAT Club?
Signature Read More
Kudos
Add Kudos
10
Bookmarks
Bookmark this Post
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 04 Sep 2026
Posts: 11,288
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,288
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
04 Dec 2020, 05:33
Bunuel
A book contains 732 pages numbered from 1 to 732. If a student randomly opens the book, what is the probability that the page number contains the digit 1?

A. 112/732

B. 184/732

C. 227/732

D. 233/732

E. 254/732

Are You Up For the Challenge: 700 Level Questions
Show more

The probability depends on numbers of integers up to 732 containing digit 1.
Let us check how many digits in each set of 100 contain 1.
Units digit as 1.....
We can have the tens digit as any of the 10 digits 0-9 => 10*1=10
Tens digit as 1......
The ones digit can be any of the digits 0-9, so => 1*10=10
Subtract 1 as the common between above two...Both include the integer 11.
Total =10+10-1=19.
a) So 0-99, 200-299, 300-399, 400-499, 500-599 and 600-699 will contain 19 integers containing 1=> 19*6=114
b) Now 100-199 will have 1 in each integer => 100
c) 700-732....701,721,731 and 710-719 => 13

Total = 114+100+13=227

Probability of any of these 227 pages to be opened = 227/732

C

_________________
Chetan Sharma,
1. Absolute modulus 2. Fractions 3. Percentage increase 4. Combinations
Signature Read More
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
Kunni
User avatar
Joined: 01 Sep 2019
Last visit: 07 Mar 2021
Posts: 29
Concentration: Marketing, Strategy
Posts: 29
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
04 Dec 2020, 14:54
chetan2u
Bunuel
A book contains 732 pages numbered from 1 to 732. If a student randomly opens the book, what is the probability that the page number contains the digit 1?

A. 112/732

B. 184/732

C. 227/732

D. 233/732

E. 254/732

Are You Up For the Challenge: 700 Level Questions

The probability depends on numbers of integers up to 732 containing digit 1.
Let us check how many digits in each set of 100 contain 1.
Units digit as 1.....10*1=10
Tens digit as 1......1*10=10
Subtract 1 as the common between above two.
Total =10+10-1=19.
a) So 0-99, 200-299, 300-399, 400-499, 500-599 and 600-699 will contain 19 integers containing 1=> 19*6=114
b) Now 100-199 will have 1 in each integer => 100
c) 700-732....701,721,731 and 710-719 => 13

Total = 114+100+13=227

Probability of any of these 227 pages to be opened = 227/732

C
Show more
Could you please tell me how did you calculate unite digit as 1 and tens digit as 1.
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 04 Sep 2026
Posts: 11,288
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,288
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
04 Dec 2020, 19:55
Kunni
chetan2u
Bunuel
A book contains 732 pages numbered from 1 to 732. If a student randomly opens the book, what is the probability that the page number contains the digit 1?

A. 112/732

B. 184/732

C. 227/732

D. 233/732

E. 254/732

Are You Up For the Challenge: 700 Level Questions

The probability depends on numbers of integers up to 732 containing digit 1.
Let us check how many digits in each set of 100 contain 1.
Units digit as 1.....10*1=10
Tens digit as 1......1*10=10
Subtract 1 as the common between above two.
Total =10+10-1=19.
a) So 0-99, 200-299, 300-399, 400-499, 500-599 and 600-699 will contain 19 integers containing 1=> 19*6=114
b) Now 100-199 will have 1 in each integer => 100
c) 700-732....701,721,731 and 710-719 => 13

Total = 114+100+13=227

Probability of any of these 227 pages to be opened = 227/732

C
Could you please tell me how did you calculate unite digit as 1 and tens digit as 1.
Show more

I have added further details in original post. Please have a look and ask if any further query.

_________________
Chetan Sharma,
1. Absolute modulus 2. Fractions 3. Percentage increase 4. Combinations
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
ShankSouljaBoi
User avatar
Joined: 21 Jun 2017
Last visit: 12 Aug 2026
Posts: 593
Location: India
Concentration: Finance, Economics
Schools: Rotman '23 (D) IIMA PGPX'22 (D) Desautels '23 (D) Schulich Sept '23 (D) WBS (D) IIML IPMX'25 (A)
GMAT 1: 660 Q49 V31
GMAT 2: 620 Q47 V30
GMAT 3: 650 Q48 V31
GPA: 3.1
WE:Corporate Finance (Non-Profit and Government)
Products:
My Reviews
Schools: Rotman '23 (D) IIMA PGPX'22 (D) Desautels '23 (D) Schulich Sept '23 (D) WBS (D) IIML IPMX'25 (A)
GMAT 3: 650 Q48 V31
Posts: 593
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
06 Dec 2020, 09:50
Hi chetan2u ,
could you please point out the flaw in my approach

Total outcomes = 732
Favourable outcomes =
repition of 1 in any three place----- no of ways = 1
+
repition of 1 only in one place----- no of ways = 3(1*9*9) = 243
+
repition of 1 in two places----- no of ways = 3(1*1*9) = 27

= 271

Final answer = 271/732 ... i am not able to figure out what cases am i actually repeating

Best
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 04 Sep 2026
Posts: 11,288
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,288
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
06 Dec 2020, 18:02
ShankSouljaBoi
Hi chetan2u ,
could you please point out the flaw in my approach

Total outcomes = 732
Favourable outcomes =
repition of 1 in any three place----- no of ways = 1
+
repition of 1 only in one place----- no of ways = 3(1*9*9) = 243
+
repition of 1 in two places----- no of ways = 3(1*1*9) = 27

= 271

Final answer = 271/732 ... i am not able to figure out what cases am i actually repeating

Best
Show more

You have found all the cases till 1000. So subtract the 1s from 732 to 1000.
19 each from 800-900 and 900-1000 and 6 from 732-800.
So 271-2*19-6=227.

_________________
Chetan Sharma,
1. Absolute modulus 2. Fractions 3. Percentage increase 4. Combinations
Signature Read More
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
carouselambra
User avatar
Current Student
Joined: 14 Mar 2018
Last visit: 28 Apr 2023
Posts: 297
Posts: 297
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
15 Dec 2020, 10:17
Hi chetansharma

Could you please help me here?
I was able to understand and work as per the the Odd/even strategy that you pointed out.
However, within abcd (Even), we can have multiple cases as shown.

Can you please point the flaw?
Thanks!
Attachments

IMG_20201215_131416.jpg
IMG_20201215_131416.jpg [ 2.73 MiB | Viewed 6410 times ]

Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 04 Sep 2026
Posts: 11,288
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,288
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
15 Dec 2020, 10:32
sharmashivng
Hi chetansharma

Could you please help me here?
I was able to understand and work as per the the Odd/even strategy that you pointed out.
However, within abcd (Even), we can have multiple cases as shown.

Can you please point the flaw?
Thanks!
Show more


Firstly in all the individual cases the denominator will be 5^5 or 3125 and not 125.
Secondly when you calculate probability of 1odd and 3even, it becomes 4C1*(2/5)^3*3/5*3/5=4*72/3125
Similarly for others

_________________
Chetan Sharma,
1. Absolute modulus 2. Fractions 3. Percentage increase 4. Combinations
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
avatar
somyamehta777
avatar
Joined: 13 Aug 2020
Last visit: 26 Sep 2024
Posts: 27
Concentration: Marketing, International Business
Posts: 27
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
15 Dec 2020, 12:55
Did this with counting technique -

1's in the hundreth digit = 1 * 10 * 10

1's in the tenth digit = 6*1*10 + 1*1*10
Here divided it into 2 parts - 1's in the tenth digit from 0-699 + 1's in the tenth digit from 700-732

1's in the unit digit = 6*9*1 + 3
Here divided it into 2 parts - 1's in units digit from 0-699 + 1's in the unit digit from 700-732(701, 711(but this is already calculated in the tenth digit), 721, 731 -> therefore, 3)

Total = 100 + 70 + 57 = 227

Probability = 227/732
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
akadiyan
User avatar
Retired Moderator
Joined: 31 May 2017
Last visit: 03 Jun 2026
Posts: 723
Concentration: Technology, Strategy
Schools: IIMA PGPX'23 Bayes Said '23
Products:
My Reviews
Schools: IIMA PGPX'23 Bayes Said '23
Posts: 723
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
15 Dec 2020, 13:07
A book contains 732 pages numbered from 1 to 732. If a student randomly opens the book, what is the probability that the page number contains the digit 1?

We can segregate this into 3 different sections

Case 1: 100 - 199 - 100 pages will have 1 in atleast one of the pages = 100

Case 2: 1 - 99 pages: 19
Page 1- 1 page / 10 pages from 10 -19 / 8 pages in unit digits in 21,31....91

The above pattern will be the same for 200s to 600s , so we have 6*19 = 114 pages

Case 3: 700 - 732
Same calculation as case 2 , but the unit digits only till 732, so we have 1+10+2 = 13 pages

Adding all the cases together we get - 100+114+13 = 227

Total outcomes: 732
Favorable outcomes: 227

Probability:\(\frac{ 227}{732}\)

Ans: C

_________________
GMATCLUB QUANT Practice Sets Link Collection |Last 30 days before GMAT | How to improve your verbal score

Profiles of Admitted applicants 2021 Intake
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
HarshavardhanR
User avatar
Joined: 16 Mar 2023
Last visit: 04 Sep 2026
Posts: 649
Status:Independent GMAT Tutor
Affiliations: Ex - Director, Subject Matter Expertise at e-GMAT
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 649
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
20 Aug 2024, 22:07
­(1) Between 1 to 9 -> 1 page contains a 1 (page 1).
(2) Between 10 to 19 -> all pages contain a 1 i.e., 19-10+1 = 10 pages.
(3) Between 20 to 99 -> 21,31...91 i.e., (91-21)/10 +1 = 8 pages.
(4) Between 100 to 199 -> all pages contain a 1 i.e., 199-100+1 = 100 pages.

So,
a) From page 1 to page 99 -> 1 + 10 + 8 = 19 pages.
b) From page 100 to 199 -> 100 pages.
c) from page 200 to 299 -> (a) repeats. The only 1's are in the ten's digit and/or one's digit i.e., 201, then 210 to 219, then 221, 231...291. Thus, there are again 19 pages.

d) Overall, from page 200 to page 699 -> 5 sets of 100 pages, each with 19 pages having 1 -> 19 x 5 = 95.

e) from 699 to 732 -> 701 (1 page), 710 to 719 (10 pages), 721 and 731 (2 pages) -> 1+10+2 = 13 pages.

Total number of pages containing 1 = (a)+(b)+(d)+(e) = 19+100+95+13 = 227 pages.

Required Probability = 227/732. Choice C.
___
Harsha
Passionate about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com

_________________
Independent GMAT Tutor | Ex-Director, Subject Matter Expertise, e-GMAT | Generalist

My website: prepsilonlearning.com
Signature Read More
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
Kinshook
User avatar
Major Poster
Joined: 03 Jun 2019
Last visit: 04 Sep 2026
Posts: 6,129
Location: India
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
My Reviews
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
Posts: 6,129
A book contains 732 pages numbered from 1 to 732. If a student randoml
Post URL
21 Aug 2024, 00:39
Given: A book contains 732 pages numbered from 1 to 732.

Asked: If a student randomly opens the book, what is the probability that the page number contains the digit 1?

Page numbers containing digit 1 = {1, 10,11,12,13,....19,21,31,41,51,61,71,81,91,100,101,110, 111,112,113,...119,120,121,..191..199,201,210,211,213,213,.219,221,231,.... 291,301,310,311, 312, ...319, 321,331,....391,401,410,411,412,413,..419,421,....491,501,510,511,512,513,...519,521,....591,601,610,611,612,613,..619,621,....691,701,710,711,712,713,..719,721,731}

Number of page numbers containing digit 1 = 2 + 9 + 8 + 100 + 2 + 9 + 8 + 2 + 9 + 8 + 2 + 9 + 8 + 2 + 9 + 8 + 2 + 9 + 8 + 2 + 9 + 2 = 19 + 100 + 19 + 19 + 19 + 19 + 19 + 13 = 227

The probability that the page number contains the digit 1 = 227/732

IMO C
­
Kudos
Add Kudos
Bookmarks
Bookmark this Post
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
Moderator:
Bunuel
Math Expert
113146 posts