i, j, and k have been declared as integers, duplicateFound has been declared to be boolean, and a has been declared to be an integer array with at least k+1 cells. duplicateFound = false; for (j = 1; j < k; ++j) for (i = k; i > j; i) if (a[i] == a[j]) duplicateFound = true; The "outer loop" is the "for" loop controlled by the loop index j . The "body of the outer loop" consists, in this example, of another loop, the "inner" (or "nested") loop controlled by the loop index i . The "body of the inner loop" in this example is an "if" statement.

(a) How many times will the body of the inner loop be executed on the first time that the body of the outer loop is executed (i.e., when j has the value 1 ) ? (Express your answer in terms of k .)

(b) How many times will the body of the inner loop be executed on the second time that the body of the outer loop is executed (when j has the value 2 ) ? (Express your answer in terms of k .)

(c) How many times will the body of the inner loop be executed on the last time that the body of the outer loop is executed (when j has the value k1 ) ?

(d) How many times altogether will the test expression a[i] == a[j] be evaluated? Express your answer as an exact polynomial in k .