Here are three “little” problems that my friend John (from UK) gave me yesterday.

$(1)$ How many 5-digit numbers can be constructed using only $1, 2$ and $3$ such that each of those three digits is used at least once?

$(2)$ Find all pairs of positive integers $(m,n)$ such that $1 + 5\cdot 2^m = n^2$.

$(3)$ Find all positive integers $a, b$ such that $a^b = b^a$.