Find number of integral solutions of $a\times b\times c\times d=210$
$$210=2\times 3\times 5\times 7$$ I tried by assuming 2,3,5,7 as numbered balls. The above problem is equivalent to placing 4 balls on 4 boxes where emplty boxes are allowed or placing 3 partitions between 4 balls. (Empty box signifies 1).
Assuming the partitions as sticks, I have to find the number of ways of arranging 4 different balls and 3 sticks. (The numbered balls between the sticks are like numbered balls in a box. So if two sticks come together, it means you get an empty box).
Number of ways = $7!$. But answer given is $8\times 4^4$
(I don't know if negative solutions are allowed. If that is the case, my method will not work. But if only positive integral solutions are allowed, is my method correct?)