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How many different positive integers exist between 10^6

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

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New post 22 Aug 2013, 11:39
3
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E

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  65% (hard)

Question Stats:

55% (01:03) correct 45% (01:23) wrong based on 186 sessions

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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

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Asif vai.....

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Joined: 02 Sep 2009
Posts: 47946
Re: How many different positive integers exist between 10^6  [#permalink]

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New post 22 Aug 2013, 11:47
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.

Answer: B.
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Senior Manager
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Joined: 10 Jul 2013
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Re: How many different positive integers exist between 10^6  [#permalink]

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New post 22 Aug 2013, 11:53
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.

Answer: B.


Great work boss.....................................................
nice..................
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Posts: 1851
Concentration: Finance
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Re: How many different positive integers exist between 10^6  [#permalink]

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New post 20 May 2014, 09:37
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.

Answer: B.


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

Thanks
Cheers
J :)
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Joined: 13 May 2014
Posts: 34
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Re: How many different positive integers exist between 10^6  [#permalink]

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New post 20 May 2014, 11:42
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.

Answer: B.


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
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Posts: 242
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Re: How many different positive integers exist between 10^6  [#permalink]

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New post 30 Jul 2014, 06:38
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.

Answer: B.


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
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Joined: 23 Jan 2013
Posts: 598
Schools: Cambridge'16
How many different positive integers exist between 10^6  [#permalink]

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New post 31 Jul 2014, 00:02
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
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Re: How many different positive integers exist between 10^6  [#permalink]

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New post 06 Jul 2018, 00:04
Hello from the GMAT Club BumpBot!

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 &nbs [#permalink] 06 Jul 2018, 00:04
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