Question

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

Answer 1

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