The number of positive numbers of not more than $10$ digits formed by $0,1,2$ and $3$ is.
$\color{green}{a.)\ 4^{10}-1} \\
b.)\ 4^{10} \\
c.)\ 4^{9}-1 \\
d.)\ \text{none of these} $
I tried ,
First digit $\rightarrow \{1,2,3\} \ 3$ digits.
Rest of digits $\rightarrow \{0,1,2,3\} \ 4$ digits.
$3\times 4^{9}$.
But the answer given in book is $a.)$
I look for a short and simple way.
I have studied maths up to $12$th grade.
Best Answer
You've missed the fact that the number could be less than 10 digits long. The number could have 9, or 8, or 7... digits.
Your working is correct, but the answer should be:
$$ 3\left(4^9 + 4^8 + \dots + 4^1 + 1\right) = 4^{10} - 1 $$