I proved a relevant theorem here ( for some reason this was closed )
Theorem 1: The number of distinct $n$-tuples of whole numbers whose components sum to a whole number $m$ is given by
$$ N^{(m)}_n \equiv \binom{m+n-1}{n-1}.$$
to apply this to your problem , you need to take into account that the theorem allows a minimum value of zero, but the dice show a minimum value of one. So you can just add one to each die so we are looking for the number of 4-tuples whose components sum to 4. I guess we are lucky that no element could possibly be greater than 6, so it will work for 6 sided dice.
The denominator is thus given by $$\binom 73 = 35 $$ which I have verified by enumeration.