Solution

Even elements in the domain are $2$ and $4$.
Even elements in the codomain are $2,4,6,8$.

The two even elements of the domain must be mapped to even elements of the codomain injectively.

Number of ways:
${}^4P_2=4\times 3=12$

Now, $2$ elements of the codomain are already used, so $6$ elements are left.

The odd elements of the domain are $1$ and $3$.
They can be mapped injectively to the remaining $6$ elements.

Number of ways:
${}^6P_2=6\times 5=30$

So,
$n=12\times 30=360$

Now,
$360=2^3\times 3^2\times 5^1$

So,
$a=3,\ b=2,\ c=1$

Therefore,
$a+b+c=3+2+1=6$