The largest number you can write down on a piece of paper, using only one digit per symbol. For example, if we have two digits to use, then 9 x 10 90 which equals 900 as our answer. But what about when there are more than 2? How many numbers could I make with 3 symbols? 4? 5? 6? etc…

I know that it’s impossible to do this problem exactly because any single number will always be larger than another but how would you approximate an upper bound for the solution? Is it possible at all?

A:

You need $\log_2$ $3^n$. This has no closed form so you should try approximating it by hand and see where it gets stuck before trying other methods.

In fact even the smallest approximation for $4^n$ goes up to around $10^{16}$, much bigger than your question suggests.

A better approach might be to note that $$\sum_{k1}^{\infty}\frac{x^k}{ !}$$ converges uniformly over $ $ to $ /e$ from below and above respectively. That means we get arbitrarily close to $900$ without needing too large a base or taking too long. &\quad 100!\cdot 0.9999