Question

Let $A=\sum _{i=1}^{10}\sum _{j=1}^{10}\min \{i,j\}$ and $B=\sum _{i=1}^{10}\sum _{j=1}^{10}\max \{i,j\}.$ Then $A+B$ is equal to ______.

Answer 1

Answer: 1100

$A=\sum _{i=1}^{10}\sum _{j=1}^{10}\min \{i,j\}$
$B=\sum _{i=1}^{10}\sum _{j=1}^{10}\max \{i,j\}$
$A=\sum _{j=1}^{10}\min (i,1)+\min (j,2)+\dots \min (i,10)$
$=\underset{19\text{times}}{\underbrace{(1+1+1+\dots +1)}}+\underset{17\text{times}}{\underbrace{(2+2+2\dots +2)}}+\underset{15\text{times}}{\underbrace{(3+3+3\dots +3)}}$
$+\dots$ (1) $1$ times
$B=\sum _{j=1}^{10}\max (i,1)+\max (j,2)+\dots \max (i,10)$
$=\underset{19\text{times}}{\underbrace{(10+10+10)}}$$+\underset{17\text{times}}{\underbrace{(9+9+9)}}$+....+11times
$\begin{array}{rl}A+B& =20(1+2+3+\dots +10)\\ & =20\times \frac{10\times 11}{2}=10\times 110=1100\end{array}$