Question

The total number of natural numbers of not more than 20 digits that are formed by using the digits 0,1, 2, 3 and 4 is

• A. $5{}^2{}^0-1$
• B. $5{}^2{}^0$
• C. $5{}^2{}^0+1$
• D. $5{}^3{}^0$

Answer 1

The correct option is A $5{}^2{}^0-1$

Number of single digit numbers are 5.
Number of two digit numbers are 4.5
Number of three digit numbers are ${4.5}^2$.
Number of 20 digit numbers are ${4.5}^{19}$
$\therefore$ Total numbers is $5+4.5+.........+{4.5}^{19}$.
$1+4+4.5+.........+{4.5}^{19}$
$=1+4(1+5+5^2+.........+5^{19})$
$=1+4\frac{5^{20}-1}{5-1}=5^{20}$
$5^{20}-1$ excluding zero (since natural numbers)