Suppose you run BB $($black box$)$ tests for some $SW.$ Your $SW$ has $5$ input categories with $3(2V+1NV),5(3V+2NV),10(6V+4NV),7(4V+3NV)$ and $8(5V+3NV)$ ECs $($equivalence classes with valid $($V$)$ and invalid $($NV$)$ input types$)$ from the first to fifth category in respective order.

a$)$ According to the BB testing hypothesis, how many test cases $n$ are required for this testing?

b$)(1.)$ Why would you prefer to represent each EC with several instead of one input? $(2.)$ From a realistic viewpoint, how does $n$

grow if several inputs instead of one test case represent each EC$?$ Hint: answer using the growing speed such as constant,

linear, cubic or exponential?

c$)$ According to the optimizing principle, what would be the minimum possible number of test cases that covers that test?

d$)$ Using the extending principle, to how many test cases would you increase the minimum number in part $($c$)$ above?