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Asifpirlo
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How many different positive integers exist between 10^6 22 Aug 2013, 11:39
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B
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Difficulty:

55% (hard)

Question Stats:

57% (01:35) correct 43% (01:41) wrong based on 344 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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B
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Bunuel
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How many different positive integers exist between 10^6 22 Aug 2013, 11:47
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Asifpirlo
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
Show more

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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Asifpirlo
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How many different positive integers exist between 10^6 22 Aug 2013, 11:53
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Bunuel
Asifpirlo
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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Great work boss.....................................................
nice..................
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How many different positive integers exist between 10^6 20 May 2014, 09:37
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Bunuel
Asifpirlo
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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Any chance we could do this with combinatorics approach instead of brute force counting?

Thanks
Cheers
J :)
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gmatacequants
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How many different positive integers exist between 10^6 20 May 2014, 11:42
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jlgdr
Bunuel
Asifpirlo
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 :)
Show more

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

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How many different positive integers exist between 10^6 31 Jul 2014, 00:02
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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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How many different positive integers exist between 10^6 18 Jan 2022, 04:39
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Method 1 to solve this GMAT Number Properties Question: Find the number of such integers existing for a lower power of 10 and extrapolate the results.
Between 10 and 100, that is 101 and 102, we have 2 numbers, 11 and 20.

Between 100 and 1000, that is 102 and 103, we have 3 numbers, 101, 110 and 200.

Therefore, between 106 and 107, one will have 7 integers whose sum will be equal to 2.

Alternative approach
All numbers between 106 and 107 will be 7 digit numbers.
There are two possibilities if the sum of the digits has to be '2'.

Possibility 1: Two of the 7 digits are 1s and the remaining 5 are 0s.
The left most digit has to be one of the 1s. That leaves us with 6 places where the second 1 can appear.
So, a total of six 7-digit numbers comprising two 1s exist, sum of whose digits is '2'.

Possibility 2: One digit is 2 and the remaining are 0s.
The only possibility is 2000000.

Total count is the sum of the counts from these two possibilities = 6 + 1 = 7
Choice B is the correct answer.
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How many different positive integers exist between 10^6 22 Feb 2026, 04:33
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