4. (20 points) Nезнayka came up with a password for his email consisting of five characters. Deciding to check the reliability of this password, he calculated all possible combinations that can be formed from these five characters. In the end, he got 20 different combinations. Is a password with this number of combinations possible, or did Nезнayka make a mistake in his calculations? If it is possible, provide an example of a suitable password.
For example, error.

Solution. The maximum number of different combinations that can be formed from 5 symbols equals $5!(n!=1 \cdot 2 \cdots n$ - the factorial of number $n$, i.e., the product of all natural numbers from 1 to $n$ inclusive). This number is obtained when all 5 symbols are distinct.

Suppose among the given 5 symbols there are $k \leqslant 5$ identical ones. Consider an arbitrary combination of these symbols. Clearly, swapping any two identical symbols in the considered combination while keeping the other symbols in their places does not yield a new combination. There can be $k!$ such permutations among the identical symbols. Thus, each combination of symbols that are not identical repeats $k!$ times. Therefore, the total number of different combinations is $5!/ k!$. Similarly, if in addition to $k$ identical symbols there are also $n$ other identical symbols, then the total number of different combinations is $5!/(k!\cdot n!)$.

It is not hard to see that 20 different combinations are obtained when $k=3, n=1$.