The concept tested in this GMAT practice question is counting method and elementary number properties.

#### Question: How many different positive integers exist between 10^{6} and 10^{7}, the sum of whose digits is equal to 2?

- 6
- 7
- 5
- 8
- 18

#### Video Explanation

Scroll for explanatory answer text#### Explanatory Answer

##### Find the number of such integers existing for a lower power of 10 and extrapolate the results.

Between 10 and 100, that is 10^{1} and 10^{2}, we have 2 numbers, 11 and 20.

Similarly, between 100 and 1000, that is 10^{2} and 10^{3}, we have 3 numbers, 101, 110 and 200.

Therefore, between 10^{6} and 10^{7}, one will have 7 integers whose sum will be equal to 2.

##### Alternative approach

All numbers between 10^{6} and 10^{7} 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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