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For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc.

In computing, a floating point operation (flop) is any one of the following operations performed by a computer: addition, subtraction, multiplication, division. For example, the dot product of two vectors $(u,v,w).(x,y,z)=ux+vy+wz$ involves $3$ multiplications and $2$ additions, a total of $5$ flops.

Calculate the exact number of flops required computing $C=AB$ for two $5\times 5$ matrices $A$ and $B$ using a direct implementation of $c_{ij}=\displaystyle\sum^5 _{k=1} a_{ik} b_{kj}$. How does this number change if both the matrices are upper triangular?