Unique Subproblems

How many unique subproblems?

Assume that 1 $\leq$ i < j $\leq$ n or 1 $\leq$ i = j $\leq$ n.

$$\left( \begin{array}{c} n \\ 2 \end{array} \right) + n$$

All ways of choosing i and j for problem m[i,j] when i < j +

All ways of choosing i and j for problem m[i,j] when i = j

$$=\; \frac{n!}{2!(n-2)!} \;+\; n$$

$$=\; \frac{n(n-1)}{2} \;+\; n$$

$$=\; n^2/2 \;-\; n/2 \;+\; n$$

$$=\; 1/2(n^2 \;+\; n)$$

$$=\; \Theta(n^2).$$

Only polynomial number of unique subproblems.