Let $S$ be the multiset $\left\{\infty \cdot e_{1}, \infty \cdot e_{2}, \infty \cdot e_{3}, \infty \cdot e_{4}\right\} .$ Determine the generating function for the sequence $h_{0}, h_{1}, h_{2}, \ldots, h_{n}, \ldots$ where $h_{n}$ is the number of $n$ -combinations of $S$ with the following added restrictions:
(a) Each $e_{i}$ occurs an odd number of times.
(b) Each $e_{i}$ occurs a multiple-of- 3 number of times.
(c) The element $e_{1}$ does not occur, and $e_{2}$ occurs at most once.
(d) The element $e_{1}$ occurs 1,3 , or 11 times, and the element $e_{2}$ occurs 2,4 , or 5 times.
(e) Each $e_{i}$ occurs at least 10 times.