Number of positive numbers of not more than $ 10$ digits formed by $0,1,2$ and $3$. 
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.
 A: 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:
$$
3\left(4^9 + 4^8 + \dots + 4^1 + 1\right) = 4^{10} - 1
$$
A: Shortcut:  You can use any of the four numbers $0, 1, 2, 3$ for any of the ten digits.  You just cannot use $0$ for all of them (because the number has to be positive).
Hence, the correct answer is $4^{10} - 1$.
A: Consider it this way:
You have 10 places that you need to fill with 4 digits. As the number can be less than 10 digits, it means that the we have no problem with filling the slots with the digit zero. Thus, each slot has 4 options: either 0 or 1 or 2 or 3.
Thus, we have $4^{10}$ options. However, as we are considering only positive numbers, one of the numbers formed by the above combinations will be $000...$ up to ten digits. As this number is not positive, we will have to subtract this case from the rest of the cases. Thus, we will have $4^{10} - 1$ total cases.
Edit: For OP only please. If you have understood my method correctly, try and think of the following approach: Suppose we look at it through the point of view of the digits. There are 4 digits with 10 possible places. So we have $10^4$ possibilities. Subtracting the 1 case as above, we have $10^4 - 1$ possibilities. However, this answer isn't the same as the previous one. Why is this so?
A: You can simply take the largest $10$-digit number on base $4$, and convert it to decimal.
This will essentially count all the numbers between $1_4$ and $3333333333_4$:
$$3333333333_4=3\sum\limits_{n=0}^{9}4^n=4^{10}-1$$
