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Senior Manager  Joined: 10 Jul 2013
Posts: 301
How many different positive integers exist between 10^6  [#permalink]

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3 00:00

Difficulty:   65% (hard)

Question Stats: 54% (01:26) correct 46% (01:38) wrong based on 198 sessions

### HideShow timer Statistics How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?

A. 6
B. 7
C. 5
D. 8
E. 18

_________________
Asif vai.....
Math Expert V
Joined: 02 Sep 2009
Posts: 56304
Re: How many different positive integers exist between 10^6  [#permalink]

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3
Asifpirlo wrote:
How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?

A. 6
B. 7
C. 5
D. 8
E. 18

So, the numbers should be from 1,000,000 to 10,000,000

The following two cases are possible for the sum of the digits to be 2:
1. Two 1's and the rest are 0's:
1,000,001
1,000,010
1,000,100
1,001,000
1,010,000
1,100,000

6 numbers.

2. One 2 and the rest are 0's:
2,000,000

1 number.

Total = 7 numbers.

_________________
Senior Manager  Joined: 10 Jul 2013
Posts: 301
Re: How many different positive integers exist between 10^6  [#permalink]

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Bunuel wrote:
Asifpirlo wrote:
How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?

A. 6
B. 7
C. 5
D. 8
E. 18

So, the numbers should be from 1,000,000 to 10,000,000

The following two cases are possible for the sum of the digits to be 2:
1. Two 1's and the rest are 0's:
1,000,001
1,000,010
1,000,100
1,001,000
1,010,000
1,100,000

6 numbers.

2. One 2 and the rest are 0's:
2,000,000

1 number.

Total = 7 numbers.

Great work boss.....................................................
nice..................
_________________
Asif vai.....
SVP  Joined: 06 Sep 2013
Posts: 1647
Concentration: Finance
Re: How many different positive integers exist between 10^6  [#permalink]

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Bunuel wrote:
Asifpirlo wrote:
How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?

A. 6
B. 7
C. 5
D. 8
E. 18

So, the numbers should be from 1,000,000 to 10,000,000

The following two cases are possible for the sum of the digits to be 2:
1. Two 1's and the rest are 0's:
1,000,001
1,000,010
1,000,100
1,001,000
1,010,000
1,100,000

6 numbers.

2. One 2 and the rest are 0's:
2,000,000

1 number.

Total = 7 numbers.

Any chance we could do this with combinatorics approach instead of brute force counting?

Thanks
Cheers
J Intern  Joined: 13 May 2014
Posts: 33
Concentration: General Management, Strategy
Re: How many different positive integers exist between 10^6  [#permalink]

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jlgdr wrote:
Bunuel wrote:
Asifpirlo wrote:
How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?

A. 6
B. 7
C. 5
D. 8
E. 18

So, the numbers should be from 1,000,000 to 10,000,000

The following two cases are possible for the sum of the digits to be 2:
1. Two 1's and the rest are 0's:
1,000,001
1,000,010
1,000,100
1,001,000
1,010,000
1,100,000

6 numbers.

2. One 2 and the rest are 0's:
2,000,000

1 number.

Total = 7 numbers.

Any chance we could do this with combinatorics approach instead of brute force counting?

Thanks
Cheers
J Hi J,

At least one part can be solved by systematic approach.
For the number to have the sum of 2, it needs two 1's as digits and rest 0's or one 2 as a digit and the rest 0's

Now for two 1's as digits and the rest 0's,
it should be b/w 10^6 and 10^7, i.e.
1 XXX,XXX -> the X designated places are to be filled by one 1 and five zeroes -> which can be done in 6 ways

Now for the number to contain 2 as a digit and the rest 0's,
it will occur when the number is 2,000,000 i.e 1 way
So, total 6+1 = 7 ways

Press kudos if you wish to appreciate
Manager  Joined: 28 Apr 2014
Posts: 205
Re: How many different positive integers exist between 10^6  [#permalink]

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jlgdr wrote:
Bunuel wrote:
Asifpirlo wrote:
How many different positive integers exist between 10^6 and 10^7, the sum of whose digits is equal to 2?

A. 6
B. 7
C. 5
D. 8
E. 18

So, the numbers should be from 1,000,000 to 10,000,000

The following two cases are possible for the sum of the digits to be 2:
1. Two 1's and the rest are 0's:
1,000,001
1,000,010
1,000,100
1,001,000
1,010,000
1,100,000

6 numbers.

2. One 2 and the rest are 0's:
2,000,000

1 number.

Total = 7 numbers.

Any chance we could do this with combinatorics approach instead of brute force counting?

Thanks
Cheers
J I think here combinatorics would be brute force method compared to simple counting
Director  G
Joined: 23 Jan 2013
Posts: 542
Schools: Cambridge'16
How many different positive integers exist between 10^6  [#permalink]

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1
I stupidly did 7!/2!*5! missing that the first 1 should be fixed. Correct way is that we have 6!/1!*5!=6 +1=7
Non-Human User Joined: 09 Sep 2013
Posts: 11720
Re: How many different positive integers exist between 10^6  [#permalink]

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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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_________________ Re: How many different positive integers exist between 10^6   [#permalink] 06 Jul 2018, 00:04
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