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

 It is currently 21 Aug 2019, 04:30

### 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

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

# How many different 3-digit integers have exactly two d

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

### Hide Tags

Manager
Joined: 30 Sep 2017
Posts: 225
Concentration: Technology, Entrepreneurship
GMAT 1: 720 Q49 V40
GPA: 3.8
WE: Engineering (Real Estate)
How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

Updated on: 03 Aug 2019, 03:56
5
2
00:00

Difficulty:

55% (hard)

Question Stats:

60% (02:06) correct 40% (02:07) wrong based on 78 sessions

### HideShow timer Statistics

How many different 3-digit integers have exactly two different digits?

(A) 1000
(B) 648
(C) 504
(D) 352
(E) 243

Give +1 kudo if you like this question

Originally posted by chondro48 on 03 Aug 2019, 03:25.
Last edited by chondro48 on 03 Aug 2019, 03:56, edited 2 times in total.
Manager
Joined: 30 Sep 2017
Posts: 225
Concentration: Technology, Entrepreneurship
GMAT 1: 720 Q49 V40
GPA: 3.8
WE: Engineering (Real Estate)
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

03 Aug 2019, 03:54
9
1
Prasannathawait wrote:
IMO the answer has to be 162.
9x9x2=162

First digit can be any among the 9 digits
Second cannot be 1 digit that is in first place, hence 9 possible options.
Third has to be one of the two digits of first or second otherwise 3 unique digits will be there.

Hi, you seems to miss the point that exactly two different digits can mean:

AAB, ABA, ABB.

I believe you missed out the third part. Have you carefully taken into account that?

AAB = 9*1*9 =81
ABA = 9*9*1 =81
ABB = 9*9*1 =81

Sum of all is 243 (E)

Hit +1 kudo if you like my solution

Posted from my mobile device
##### General Discussion
Manager
Joined: 10 Aug 2018
Posts: 213
Location: India
Concentration: Strategy, Operations
WE: Operations (Energy and Utilities)
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

03 Aug 2019, 03:42
IMO the answer has to be 162.
9x9x2=162

First digit can be any among the 9 digits
Second cannot be 1 digit that is in first place, hence 9 possible options.
Third has to be one of the two digits of first or second otherwise 3 unique digits will be there.
_________________
On the way to conquer the GMAT and I will not leave it until I win. WHATEVER IT TAKES.
Target 720+

" I CAN AND I WILL"

1) Gmat prep: 620 Q48, V27
2) Gmat prep: 610 Q47, V28
3) Gmat prep: 620 Q47, V28
4) Gmat prep: 660 Q47, V34
5) Gmat prep: 560 Q37, V29
6) Gmat prep: 540 Q39, V26
7) Veritas Cat: 620 Q46, V30
8) Veritas Cat: 630 Q45, V32
Senior Manager
Joined: 16 Jan 2019
Posts: 412
Location: India
Concentration: General Management
WE: Sales (Other)
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

03 Aug 2019, 03:51
5
1
Number of 3-digit integers that have exactly two different digits = Total number of 3 digit integers $$-$$ Number of 3 digit integers with 0 different digits (All digits are same) $$-$$ Number of 3 digit integers with 3 different digits (All digits are different)

Total number of 3 digit integers = 999-100+1 = 900

Number of 3 digit integers with 0 different digits (All digits are same) = 9*1*1 = 9 (These are 111, 222, 333 ..... 999)

Number of 3 digit integers with 3 different digits (All digits are different) = 9*9*8 = 648 (First digit can be 1-9, second digit can be 0-9 except the one used for first digit, third digit can be 0-9 except the digits used for first and second digits)

So, Number of 3-digit integers that have exactly two different digits = 900-9-648 = 243

Manager
Joined: 28 Jan 2019
Posts: 118
Location: Peru
How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

Updated on: 03 Aug 2019, 04:11
2
We can translate different 3-digit integers have exactly two different digits as this:

All 3 digit integers - all 3 different digit integers - all 3 same digit integers = all 3 digit integers with two different digits, so:

All 3 digit integers = 9 * 10 * 10
all 3 different digit integers = 9 * 9 * 8
all 3 same digit integers = 111, 222, 333... = 9

all 3 digit integers with two different digits = 900 - 648 - 9 = 243

(E) is our answer

Originally posted by Mizar18 on 03 Aug 2019, 04:01.
Last edited by Mizar18 on 03 Aug 2019, 04:11, edited 1 time in total.
Manager
Joined: 30 Sep 2017
Posts: 225
Concentration: Technology, Entrepreneurship
GMAT 1: 720 Q49 V40
GPA: 3.8
WE: Engineering (Real Estate)
How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

Updated on: 03 Aug 2019, 08:57
Anyone, try this question with timer, leave out good explanation, and kindly leave +1 kudo if this question is quite good.

Originally posted by chondro48 on 03 Aug 2019, 04:04.
Last edited by chondro48 on 03 Aug 2019, 08:57, edited 3 times in total.
Manager
Joined: 10 Aug 2018
Posts: 213
Location: India
Concentration: Strategy, Operations
WE: Operations (Energy and Utilities)
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

03 Aug 2019, 04:12
I missed the question completely.

chondro48 wrote:
Prasannathawait wrote:
IMO the answer has to be 162.
9x9x2=162

First digit can be any among the 9 digits
Second cannot be 1 digit that is in first place, hence 9 possible options.
Third has to be one of the two digits of first or second otherwise 3 unique digits will be there.

Hi, you seems to miss the point that exactly two different digits can mean:

AAB, ABA, ABB.

I believe you missed out the third part. Have you carefully taken into account that?

AAB = 9*1*9 =81
ABA = 9*9*1 =81
ABB = 9*9*1 =81

Sum of all is 243 (E)

Hit +1 kudo if you like my solution

Posted from my mobile device

_________________
On the way to conquer the GMAT and I will not leave it until I win. WHATEVER IT TAKES.
Target 720+

" I CAN AND I WILL"

1) Gmat prep: 620 Q48, V27
2) Gmat prep: 610 Q47, V28
3) Gmat prep: 620 Q47, V28
4) Gmat prep: 660 Q47, V34
5) Gmat prep: 560 Q37, V29
6) Gmat prep: 540 Q39, V26
7) Veritas Cat: 620 Q46, V30
8) Veritas Cat: 630 Q45, V32
Senior Manager
Joined: 24 Nov 2016
Posts: 324
Location: United States
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

03 Aug 2019, 08:42
1
chondro48 wrote:
exc4libur, try this question and leave +1 kudo if this question is quite good.

[method 1]
three-digit numbers with 2 different digits has 3 cases:
XXY: hundreds ≠ 0 so 9 choices • tens = hundreds = 1 choice • units ≠ other digits = 10-1 = 9 choices … 9*1*9=81
XYX: hundreds ≠ 0 so 9 choices • units = hundreds = 1 choice • tens ≠ other digits = 10-1 = 9 choices … 9*1*9=81
YXX: hundreds ≠ 0 so 9 choices • tens ≠ hundreds = 9 choices • units = hundreds = 1 choice … 9*9*1=81

total: 81+81+81=243

[method 2]
3d numbers - 3d triplets - 3d's with different digits = 3d numbers with 2 different digits:
3d numbers = 9*10*10 = 900 or (range between 999-100 inclusive is 999-100+1=900)
3d triplets = 9*1*1 (9 because hundreds ≠ 0) = 9
3d's with different digits = 9*9*8= 81*8 = 648
total: 900-9-648=243

Intern
Joined: 13 Apr 2019
Posts: 43
Location: India
Concentration: Marketing, Operations
GPA: 3.5
WE: General Management (Retail)
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

07 Aug 2019, 22:16
chondro48 wrote:
How many different 3-digit integers have exactly two different digits?

(A) 1000
(B) 648
(C) 504
(D) 352
(E) 243

Give +1 kudo if you like this question

Approach: Total - unfavourable cases with zero

Total: 10C2*2C1*3

1) 1 zero and 2 any other number (first digit can't be zero, everything else is acceptable): 9
2) 2 zero and 1 any other number (other number has to be in hundredth place, everything else is not acceptable): 9*2

Therefore answer: 10C2*2C1*3 - (9 +9*2)
243
Intern
Joined: 10 Mar 2018
Posts: 6
Location: India
Concentration: Entrepreneurship, Marketing
Schools: ISB '21, IIMB, IIM
WE: Design (Retail)
Re: How many different 3-digit integers have exactly two d  [#permalink]

### Show Tags

08 Aug 2019, 01:31
chondro48 wrote:
Prasannathawait wrote:
IMO the answer has to be 162.
9x9x2=162

First digit can be any among the 9 digits
Second cannot be 1 digit that is in first place, hence 9 possible options.
Third has to be one of the two digits of first or second otherwise 3 unique digits will be there.

Hi, you seems to miss the point that exactly two different digits can mean:

AAB, ABA, ABB.

I believe you missed out the third part. Have you carefully taken into account that?

AAB = 9*1*9 =81
ABA = 9*9*1 =81
ABB = 9*9*1 =81

Sum of all is 243 (E)

Hit +1 kudo if you like my solution

Posted from my mobile device

Hi,
I solved the same way, but I took one additional case apart from the three cases (AAB, ABA, ABB) and that is BAA.
For example;
AAB= 223
ABA= 232
ABB= 233
BAA= 322
Can you please explain where did I go wrong?
_________________
~ETERNAL~
Re: How many different 3-digit integers have exactly two d   [#permalink] 08 Aug 2019, 01:31
Display posts from previous: Sort by

# How many different 3-digit integers have exactly two d

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

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