3 primes problem

Given prime number $p$, how many triples of distinct prime numbers $(a,b,c)$ are there for a such that $a^a + b^b + c^c$ is a multiple of $p$?

According to diriclet theorem There exist infinite many odd primes for $b$ and $c$ such that $b \equiv 1 \pmod p$ and $c \equiv -1 \pmod p$. If we take $a$ to be $p$, then we get:
$\begin{aligned}a^a+b^b+c^c &\equiv 0^a+1^b+(-1)^c &\pmod p \\&\equiv 0+1+(-1) &\pmod p \\&\equiv 0&\pmod p \\\end{aligned}$
So $a^a + b^b + c^c$ is a multiple of $p$. As there exist infinitly many values for $b$ and $c$, there are infinitly many triples.