Question

Solve the following question and mark the best possible option.
Using the digits 1,2 find the number of natural numbers, which are smaller than 2 × 10⁸.

Options

• 512

• 255

• 256

• 766

Answer 1

The number can contain maximum of 9 digits and for every place, there are 2 choices either 1 or 2.
If no. is 1-digited = 2 nos. are possible
If no. is 2-digited = 2² nos. are possible
If no. is 3-digited = 2³ nos. are possible
If no. is 4-digited = 2⁴ nos. are possible
If no. is 5-digited = 2⁵ nos. are possible
If no. is 6-digited = 2⁶ nos. are possible
If no. is 7-digited = 2⁷ nos. are possible
If no. is 8-digited = 2⁸ nos. are possible
If no. is 9-digited = first place can be filled in the way i.e. by taking 1. Reaming 8 places can be filled either by 1 or by 2 i.e. 2 choices. So, 1 x 2⁸ nos. are possible in this case.
So, required answer = (2¹ + 2² + ------ + 2⁸) + 2⁸ = 766.