I suddenly thought of a question today: What is the best $n$-digit password? It is not specific so I'll write it in a better way:
There is a password lock that has $n$ digits. There are $t$ choices for every digit. There is a thief that wants to crack the password lock, so he blows some powder into the lock that will show the fingerprints and will tell him the digits used (If there are repeated digit in the password, it only shows one fingerprint on the repeated digit). If the password consists of $m$ distinct digits, then find $m$ ($m\le n$) that makes the number of the combination of the possible password $P\left(m\right)$ the most.
Let me show a example:
$\therefore m=3$ is the answer for the case $n=4,t=4$.
However, when $n,t$ are bigger number, it will be hard to calculate. Hence, I want to ask you guys the general case or making a table. Thank you!