Question

A full-scale reinforced concrete building was designed and tested under simulated earthquake loading conditions (Journal of Structural Engineering, Jan. 1986). After completion of the experiments, several design engineers were administered a questionnaire in which they were asked to evaluate two building parameters (size and reinforcement) for each of three parts (shear wall, columns, and girders). For each parameter–part combination, the design engineers were asked to choose one of the following three responses: too heavy, about right, and too light.

1. How many different responses are possible on the questionnaire?

2. Suppose the design engineers are also asked to select the three parameter–part combinations with the overall highest ratings and rank them from 1 to 3. How many different rankings are possible?

Answer 1

There are 2 building parameters for each of 3 parts of a building.

Thus there are 3$\times$2 = 6 parameter-part combinations.

For each parameter–part combination 3 responses are possible.

Total number of different responses possible = 3$\times$6 = 18

.

3 parameter–part combinations will have the overall highest ratings among the 6 parameter–part combinations in ⁶C₃ = 20 ways. These 3 parameter–part combinations will have rank from 1 to 3 in 3! = 6 permutations.

Thus total number of different rankings possible = 20$\times$6 = 120