LetA$[1...$ n$]$ be an array of n distinct numbers. If i $<$ j and A$[$i$]>$ A$[$j$],$ then the pair

$($i$,$ j$)$ is called an inversion of A$.$

$($a$)$ What is the maximum number of inversions an array A can have?

$($b$)$ What is the minimum number of inversions an array A can have?

$($c$)$ determine the inversions of A $=<3,16,1,5,8,4,7>$ list them all

$($d$)$ provided modify the insertion$-$sort algorithm so that the algorithm counts the

total number of inversions

$($e$)$ Suppose A$[1...$ n$]$ and B$[1...$ n$]$ are two arrays with identical elements $($not neces$-$

sarily same order$).$ Professor Stooge has hypothesized that if these two arrays have the

same number of inversions then the order of the arrays are identical. Is Professor Stooge

correct? Explain.