Principle of I/E: In how many ways can eight cakes be distributed among four kids if every kid receives at least one cake?

Question

Eight cakes are distributed randomly among four kids. Use I/E to determine in how many of the possible distributions every kid receives at least $1$ cake.

Hint: Define $A_i$ to be the set of distributions where the $i$th kid does not receive any cake. Compute $|\cap_{i \in I} A_i|$ if $|I| = k \leq 4$.