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How many positive, 3-digit integers contain at least one 7?

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BrentGMATPrepNow
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How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   Updated on: 14 May 2024, 12:36
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How many positive, 3-digit integers contain at least one 7?

A) 171
B) 200
C) 243
D) 252
E) 271

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Here's our video solution: https://www.gmatprepnow.com/module/gmat ... /video/784­
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D

Originally posted by BrentGMATPrepNow on 24 Oct 2016, 09:39.
Last edited by Bunuel on 14 May 2024, 12:36, edited 2 times in total.
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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   16 Aug 2019, 06:40
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GMATPrepNow wrote:
How many positive, 3-digit integers contain at least one 7?

A) 171
B) 200
C) 243
D) 252
E) 271

NOTE: Kudos for all correct solutions

Here's our video solution: https://www.gmatprepnow.com/module/gmat ... /video/784


finding those with at least a 7 (XX7,X77,777) is the same as finding those without a 7 and subtracting it from the total
num of digits: 0 to 9 = 10
3-digit positive numbers: 999-100+1=900
first digit ≠ 0 or 7: 8 options
second digit ≠ 7: 9 options
third digit ≠ 7: 9 options
total 3-digit num ≠ 7: 8*9*9=648
total 3-digit with at least a 7: total - no 7 = 900 - 648 = 252

Answer (D).
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acegmat123
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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   Updated on: 24 Oct 2016, 10:51
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Contains only one 7:

7XX: 1*9*9 = 81

X7X: 8*1*9 = 72

XX7: 8*9*1 = 72

Contains two 7s:

77X: 1*1*9=9

X77: 8*1*1 = 8

Contains three 7s:

777: 1

Total 243

C

Edited: Editing as per GMATPrep's comments


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Originally posted by acegmat123 on 24 Oct 2016, 10:13.
Last edited by acegmat123 on 24 Oct 2016, 10:51, edited 1 time in total.
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BrentGMATPrepNow
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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   24 Oct 2016, 10:19
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acegmat123 wrote:
Contains only one 7:

7XX: 1*10*10= 100

X7X: 9*1*10 = 90

XX7: 9*10*1 = 90

Contains two 7s:

77X: 1*1*10= 10

X77: 9*1*1 = 9

Contains three 7s:

777: 1

Total 200

B


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Be careful. You have....
Contains only one 7:

7XX: 1*10*10= 100

This configuration allows for more than one 7 since we're saying that the 2nd and 3rd digits can be ANY digit (including 7)
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acegmat123
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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   24 Oct 2016, 10:52
GMATPrepNow wrote:
acegmat123 wrote:
Contains only one 7:

7XX: 1*10*10= 100

X7X: 9*1*10 = 90

XX7: 9*10*1 = 90

Contains two 7s:

77X: 1*1*10= 10

X77: 9*1*1 = 9

Contains three 7s:

777: 1

Total 200

B


Sent from my iPhone using GMAT Club Forum mobile app


Be careful. You have....
Contains only one 7:

7XX: 1*10*10= 100

This configuration allows for more than one 7 since we're saying that the 2nd and 3rd digits can be ANY digit (including 7)


Thanks GMATPerpNow. Completely overlooked that part.


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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   24 Oct 2016, 11:08
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acegmat123 wrote:
Contains two 7s:

77X: 1*1*9=9

X77: 8*1*1 = 8



There can be one more combination

7x7 : 1*9*1 = 9

So, the Total possibility wil be 243+9 = 252

Answer will (D)

PS : What say Brent ?
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BrentGMATPrepNow
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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   24 Oct 2016, 11:44
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Abhishek009 wrote:
acegmat123 wrote:
Contains two 7s:

77X: 1*1*9=9

X77: 8*1*1 = 8



There can be one more combination

7x7 : 1*9*1 = 9

So, the Total possibility wil be 243+9 = 252

Answer will (D)

PS : What say Brent ?


Bingo!
As we can see, accounting for every possible scenario is much more time-consuming than taking the TOTAL number of possible outcomes (that both adhere and break the rule) and then subtract the number of outcomes that break the rule.

Cheers,
Brent
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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   24 Oct 2016, 12:39
GMATPrepNow wrote:
Abhishek009 wrote:
acegmat123 wrote:
Contains two 7s:

77X: 1*1*9=9

X77: 8*1*1 = 8



There can be one more combination

7x7 : 1*9*1 = 9

So, the Total possibility wil be 243+9 = 252

Answer will (D)

PS : What say Brent ?


Bingo!
As we can see, accounting for every possible scenario is much more time-consuming than taking the TOTAL number of possible outcomes (that both adhere and break the rule) and then subtract the number of outcomes that break the rule.

Cheers,
Brent


Yes you're right. Learnt a new method today.

9*10*10 - 8*9*9 = 900-648 = 252


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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   16 Aug 2019, 07:18
Let's calculate the numbers which don't have any 7 in their digits.
8×9×9=648
There are 900 numbers totally between 100 and 999
900-648=252
Option D

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Re: How many positive, 3-digit integers contain at least one 7? [#permalink] How many positive, 3-digit integers contain at least one 7?   14 May 2024, 12:25
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