chetan2u wrote:

How many different positive numbers smaller than\(2*10^8\) can be formed using the digits 1 and 2 only?

A. 256

B. 510

C. 512

D. 766

E. 6561

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

.

.

.

.

.

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

.

.

.

.

.

[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 =

2^8If 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

--------------------------

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:

http://www.gmatprepnow.com/module/gmat-counting/video/775You can also watch a demonstration of the FCP in action:

https://www.gmatprepnow.com/module/gmat ... /video/776 Cheers,

Brent

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