Answer
(a) $479001600$
(b) $95040$
(c) $3326400$
Work Step by Step
(a) Here, all $12$ components are different. We can make a design by arranging these components into a cylindrical casing.
So there are $12!=479001600$ designs possible.
(b) Here, $7$ components are identical, and the other $5$ are different.
So there are $\frac{12!}{7!1!1!1!1!1!}=95040$ designs possible.
(c) Here, $3$ components are identical to one type, $4$ components are similar to second type, and rest $5$ components are different from each other.
So there are $\frac{12!}{3!4!1!1!1!1!1!}=3326400$ designs possible.
