Question

The number of natural numbers which are less than 2 . 10⁸ and which can be written by means of the digit 1 and 2 is –

• A. 772
• B. 870
• C. 900
• D. 766

Answer 1

Answer: (D)

766

Answer 2

The required numbers are

1, 2, 11, 12, 21, 22, …. , 122222222

Let us calculate how many numbers are these.

There are 2 one-digit such numbers. There are 22 two-digit such numbers and so on.

There are 2⁸ eight-digit such numbers. All the digit numbers beginning with 1 and written by means of 1 and 2 are smaller than 2. 10⁸. Thus, there are 2⁸ such nine-digit numbers.

Hence the required number of numbers is

2 + 2² + 2³ + …. + 2⁸ + 2⁸

= $\frac{2(2^{8} - 1)}{2 - 1}$+ 2⁸ = 2⁹ – 2 + 2⁸ = 766.