In 8 and 9, how many times will the innermost loop be iterated when the algorithm segment is implemented and run? Assume n, m, k, and j are positive integers.

8. for m :   = 1 to n
for k :   = 1 to m
for j :   = 1 to k
for i :   = 1 to j
[Statements in the body of the inner loop, none containing branching statements that lead outside loop]
next i
next j
next k
next m

To find the number of times that the innermost loop be iterated when the algorithm segment is implemented and run. Assume n, m, k and j are positive integers.

for m:=1 to n     for k:=1 to m          for j:=1 to k              for i:=1 to j                   [Statement in the body of the inner loop, none containing branching statements that lead outside the loop]              next i          next j     next knext m

Given information:

Construct a trace table for the values of m,k,j and i for which the statement in the body of the inner most loop are executed.

Concept used:

Observe that i goes from 1 to j. It is always the case that i≤j. Similarly, because j goes from 1 to k, it is always the case that j≤k and similarly k≤m.

Observe that there is one iteration of the inner most loop for each column and there is one column of the table for each quadruples of integers (i,j,k,m) with 1≤i≤j≤k≤m≤n.

Calculation:

The total number of quadruples is same as 4 combinations with repetition allowed that can be formed from a set of n elements