How many different positive numbers smaller than 2*10^8 can be formed

Let's start with all 9-digit numbers that are less than 200,000,000
So, let's take the task of creating 9-digit numbers and break it into stages.

Stage 1: Select the 1st digit.
This digit must be 1, so we can complete stage 1 in 1 way

Stage 2: Select the 2nd digit.
This digit can be either 1 or 2, so we can complete stage 2 in 2 ways

Stage 3: Select the 3rd digit.
This digit can be either 1 or 2, so we can complete stage 3 in 2 ways

Stage 4: Select the 4th digit.
This digit can be either 1 or 2, so we can complete stage 4 in 2 ways

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Stage 9: Select the 9th digit.
This digit can be either 1 or 2, so we can complete stage 3 in 2 ways

By the Fundamental Counting Principle (FCP), we can complete all 9 stages (and thus create a 9-digit number) in (1)(2)(2)(2)(2)(2)(2)(2)(2) ways
This equals 2^8.



Now, we'll count all 8-digit numbers
So, let's take the task of creating 8-digit numbers and break it into stages.

Stage 1: Select the 1st digit.
This digit can be 1 or 2 , so we can complete stage 1 in 2 ways

Stage 2: Select the 2nd digit.
This digit can be either 1 or 2, so we can complete stage 2 in 2 ways
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[u]Stage [8/u]: Select the 8th digit.
This digit can be either 1 or 2, so we can complete stage 3 in 2 ways

By the FCP, the total number of outcomes = 2^8


If we follow the same steps for 7-digit numbers, we get a total of 2^7 outcomes.

And so on.

So, the TOTAL number of possibilities = 2^8 + 2^8 + 2^7 + 2^6 ..... + 2^1 = 766

Answer: D
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Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting/video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Cheers,
Brent