Question

In the design of an electromechanical product, 12 components are to be stacked into a cylindrical casing in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom & the other end is the top.
a) If all components are different, how many different designs are possible?
b) If 7 components are the same, & others are different, how many different designs are possible?
c) If 3 components are of one type, and 4 of another, and other 5 are different, how many different designs are possible?

Answer 1

For 12 different components the rule for permutations (all possible orderings) is n! = n(n-1)(n-2)...

The other questions are too hard for my brain this morning.

Answer 2

b) 12!/7! (since 7! are repeated)
c) 12!/(3! 4! 5!) ... divide by repeated.

Answer 3

for (b) I worried that the placement of the 7 within the total of twelve added more possibilities than just permuting the order of the 7.
Similar complexity for (c) I thought.

Answer 4

@experimentX , in part be you did not consider presence of 5 different components ..
and for part (c) there are 3 one type and 4 another type but remaining last 5s are different how can we assume same solution for identical and different components

Answer 5

that's a bad way to go .. it makes problem more complicated.
just think that you are permuting 12 objects (which includes 5 as well as 7)

and in 7 (identical object) think about the permutation there. this is again counted on 12!

Answer 6

total permutation = permutation with identical objects * permutation of spaces occupied by identical objects.