Question

Number of different natural numbers which are smaller than two hundred million & using only the digit $1$ or $2$ is :

• A. $3{.2}^8-2$
• B. $\left(3\right){.2}^8-1$
• C. $\left(3\right){.2}^9-1$
• D. none

Answer 1

The correct option is A $3{.2}^8-2$

Number of different natural number smaller than two hundred million using only digit $1$ or $2$ will be either of single digit or 2 digit or $9$ digit numbers using either 1 or 2

This number can be given as

$N=2^8+2^8+2^7+2^6+2^{5+}+2^4+2^3+2^2+2^1$

$=2^8+{\sum }_{i=1}^8{2}^i$

$=2^8+2{\sum }_{i=0}^7{2}^i$ and for sum of ${\sum }_{i=0}^7{2}^i$ we have

${\sum }_{i=0}^7{2}^i=\frac{1-2^8}{1-2}=2^8-1$

$=2^8+2(2^8-1)$

Hence the number of such numbers is $3.2^8-2$

option $\left(A\right)$ none