[Solved]: Analyzing programs with multiple for-loops

From your comments, it seems that you are incorrectly calculating the number of loops. And since I do not where to start from for you, I will start from a basic level – please ignore whatever you know or find obvious. I am also going to assume that the semi-colons after each for loop are not present – otherwise in many programming languages, the variable x will be incremented only twice, once after each apparently nested loops. Counting how many times each loop executes can be written as a discrete summation. For example, a simple for loop from $1$ to $n$, where $1$ operation takes place, can be seen as $sumlimits_{i = 1}^{n}1 = n$. That is, the loop will execute $n$ times (or the increment statement will be executed $n$ times). Nested loops can be counted as nested summations. When evaluating any summation, consider the iterating variable as the variable, and treat everything else as a constant. Consider the first set of nested loops. The number of executions/number of times x is incremented can be written as: $$T_1(n) = sumlimits_{i = 1}^{n}sumlimits_{j = i}^{n}sumlimits_{k = 1}^{in}1$$ Notice the upper and lower limits on each summation, and compare with your loop variables and ranges. We start evaluation from the innermost loop as follows: $$begin{align*} T_1(n) & = sumlimits_{i = 1}^{n}sumlimits_{j = i}^{n}sumlimits_{k = 1}^{in}1 = sumlimits_{i = 1}^{n}sumlimits_{j = i}^{n}in &= sumlimits_{i = 1}^{n}left(in sumlimits_{j = i}^{n}1right) text{(considering $j$ as the only variable)} &= sumlimits_{i = 1}^{n} left(in(n – i + 1) right) = sumlimits_{i = 1}^{n}((n^2 + n)i – ni^2) &= sumlimits_{i = 1}^{n}left((n^2 + n)iright) – sumlimits_{i = 1}^{n}left(ni^2right) &= (n^2 + n)sumlimits_{i = 1}^{n}i – nsumlimits_{i = 1}^{n}i^2 &= n(n+1)cdot frac{n(n+1)}{2} – n cdot frac{n(n+1)(2n+1)}{6} &= frac{n^2(n+1)(n+2)}{6} end{align*} $$ You can verify that if x is initially 0, this will be its value after the first set of nested loops. For the second set, I will provide the the summation and answer, leaving the actual work up to you. $$T_2(n) = sumlimits_{i = 1}^{n-1}sumlimits_{j = i+1}^{n}sumlimits_{k = 1}^{j}1 = frac{n(n-1)(n+1)}{3}$$ As others have pointed out, since the two blocks of nested loops are serial, you can just add the number of times each has executed to get the final count as $T(n) = T_1(n) + T_2(n)$. This will give you the number of times x has been incremented. This simple mechanical process should help you out in most instances of nested loops.