NIMCET 2018 — Mathematics PYQ

Concept:

The total number of natural numbers which are smaller than $2\times 10^8$ and vhich contains only the digits 1 and 2 is the sum of natural numbers which contains only the digits 1 and 2 of 1 digit,2 digit ,3 digit ,4 digit, 5 $\mathrm{digit},6\mathrm{digit},7\mathrm{digit},8,\mathrm{digit},9digit.$

Calculations

Given Number is $2\times 10^8=20,00,00,000$ i.e 9 digit number.
To find the total number of natural numbers which are smaller than $2\times 10^8$ and which contains only the digits 1 and 2 is the sum of natural numbers which contains only the digits 1 and 2 of 1 digit, 2 digit , 3 digit , 4 digit, 5 digit , 6 digit , 7 digit , 8, digit. 9 digit. 
Natural numbers which contains only the digits 1 and 2 of 1 digit $=1,2=2^1$
Natural numbers which contains only the digits 1 and 2 of 2 digits $=11,12,21,22=4=2^2$
Natural numbers which contains only the digits 1 and 2 of 3 digits =111,112,121,122,222,221,212,212,211=8
$=2^3$

continue in this way

Natural numbers which contains only the digits 1 and 2 of 4 digits $=2^4$

Natural numbers which contains only the digits 1 and 2 of 5 digits $=2^5$

Natural numbers which contains only the digits 1 and 2 of 6 digits $=2^6$

Natural numbers which contains only the digits 1 and 2 of 7 digits $=2^7$

 

Natural numbers which contains only the digits 1 and 2 of 8 digits = $2^8$

Natural numbers which contains only the digits 1 and 2 of 9 digit i.e. 20,00,00,000 = $\frac{2^9}{2}=2^8$

Now,

The total number of natural numbers which are smaller than $2\times 10^8$ and which contains only the digits 1 and 2 = $1+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^8$

= 766