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 and the other end is the top. (a) if all components are different, how many different designs are possible? (b) if seven components are identical to one another, but the others are different, how many different designs are possible? (c) if three components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible?

(a) 479,001,600 combinations (b) 95,040 combinations (c) 3,326,400  combinations

Step-by-step explanation:

(a) In the case we have 12 different elements to stack, the possible combinations can be calculated as a premutation of 12 elements:

P = 12! = 479,001,600 combinations

(b) If seven components are identical, we have a permutation with repetition.

We have 7!=5040 combinations that are the same, so there is only one unique out of 5040 combinations. We have to divide the total possible combinations by 5040.

P = 12! / 7! = 479,001,600 / 5,040 = 95,040 combinations

(c) In the same way, if we have 3 components of Type 1 and 4 components of Type 2 (and the others all different), we have 3!*4! = 144 combinations that are the same.

P = 12!/(3!*4!) = 479,001,600 / 144 = 3,326,400  combinations

translate k to n and reflect across the line containing jk.