## Question

The number of natural numbers which are smaller than 2 × 10^{8} and which can be written by means of the digits and 2 is ………..

### Solution

766

The required number 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 2^{2} two digit such numbers. And so on.

There are 2^{8} eight digit such numbers. All the nine digit numbers beginning with 1 and written by means of 1 and 2 are smaller than 2.10^{8}. Thus, there are 2^{8} such nine digit numbers.

Hence, the required number of numbers is

.

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