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Of the three-digit integers greater than 660, how many have

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Of the three-digit integers greater than 660, how many have [#permalink] New post 10 Mar 2012, 04:45
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30% (02:47) correct 70% (01:30) wrong based on 1 sessions
Of the three-digit integers greater than 660, how many have two digits that are equal to each other and the remaining digit different from the other two?
(A) 47
(B) 60
(C) 92
(D) 95
(E) 96

I always get stuck on these questions. Any idea what is the concept and how can I solve?
[Reveal] Spoiler: OA

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Re: 3 digit integers [#permalink] New post 10 Mar 2012, 05:36
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enigma123 wrote:
Of the three-digit integers greater than 660, how many have two digits that are equal to each other and the remaining digit different from the other two?
(A) 47
(B) 60
(C) 92
(D) 95
(E) 96

I always get stuck on these questions. Any idea what is the concept and how can I solve?


Three digit number can have only following 3 patterns:
A. all digits are distinct;
B. all three digits are alike;
C. two digits are alike and third digit is different.

We need to calculate C. C = Total - A - B

Total numbers from 660 to 999 = 339 (3-digit numbers greater than 660);
A. all digits are distinct = 3*8+3*9*8 = 240. If first digit is 6 then second digit can take only 3 values (7, 8, or 9) and third digit can take 8 values. If first digit is 7, 8, or 9 (3 values) then second and third digits can take 9 and 8 values respectively;
B. all three digits are alike = 4 (666, 777, 888, 999).

So, 339-240-4=95.

Answer: D.
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Re: Of the three-digit integers greater than 660, how many have [#permalink] New post 10 Mar 2012, 09:40
my way of thinking is not elegant :) but anyways , see what I did -
we have the range 661-998

1. two 1st digits are the same
then we have-
661-669 (8 options (exclude 666))
770-779(9 options (exclude 777))
880-889 (9 options (exclude 888))
990-998 (9 options (exclude 999))
total=9*3+8=35

2.the 1st and the last are the same
then we have
6*6 (3 options- 676 686 696))
7*7 (9 options (exclude 777))
8*8 (9 options (exclude 888))
9*9 (9 options (exclude 999))
total=9*3+3=30

3. the 2nd and the 3d terms are the same.
then we have
66x- none option, since our first two digits are already the same, and the 3d cant be the same
7xx -9 options (exclude 777)
8xx- 9 options (exclude 888)
9xx 9 options (exclude 999)
total=9*3=27

30+35+27=92

cant find the rest 3 numbers to get 95. I will appreciate if someone helps me :)
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Re: Of the three-digit integers greater than 660, how many have [#permalink] New post 10 Mar 2012, 09:53
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LalaB wrote:
my way of thinking is not elegant :) but anyways , see what I did -
we have the range 661-998

1. two 1st digits are the same
then we have-
661-669 (8 options (exclude 666))
770-779(9 options (exclude 777))
880-889 (9 options (exclude 888))
990-998 (9 options (exclude 999))
total=9*3+8=35

2.the 1st and the last are the same
then we have
6*6 (3 options- 676 686 696))
7*7 (9 options (exclude 777))
8*8 (9 options (exclude 888))
9*9 (9 options (exclude 999))
total=9*3+3=30

3. the 2nd and the 3d terms are the same.
then we have
66x- none option, since our first two digits are already the same, and the 3d cant be the same
7xx -9 options (exclude 777)
8xx- 9 options (exclude 888)
9xx 9 options (exclude 999)
total=9*3=27

30+35+27=92

cant find the rest 3 numbers to get 95. I will appreciate if someone helps me :)


You are missing 677, 688 and 699.
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PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


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Re: Of the three-digit integers greater than 660, how many have [#permalink] New post 10 Mar 2012, 09:56
gosh, only I can do such stupid mistakes! all efforts are in vain, if u come to the wrong answer at last.
thnx Bunuel. appreciate ur help. next time I will be more attentive (at least I hope so)
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Re: Of the three-digit integers greater than 660, how many have   [#permalink] 10 Mar 2012, 09:56
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