An integer container (x, y, z) is a rectangular prism with positive integer side lengths x, y, z, where x \leq y \leq z. A stick has x=y=1; a flat has x=1 and y>1; and a box has x>1. There are 5 integer containers with volume 30: one stick (1,1,30), three flats (1,2,15),(1, 3,10),(1,5,6) and one box (2,3,5).
a. How many sticks, flats and boxes are there among the integer containers with volume 36?
b. How many flats and boxes are there among the integer containers with volume 210?
c. Suppose n=p_{1}^{e_{1}} \cdots p_{k}^{e_{k}} has k distinct prime factors p_{1}, p_{2}, \ldots, p_{k}, each with integer exponent e_{1} \geq 1, e_{2} \geq 1, \ldots, e_{k} \geq 1 and k \geq 3. How many boxes are there among the integer containers with volume n ? Express your answer in terms of e_{1}, e_{2}, \ldots, e_{k}. How many boxes with volume n=8! are there?