The value of $$\sum\limits_{n = 1}^{10} {\sum\limits_{m = 1}^{10} {\left( {{m^2} + {n^2}} \right){\text{equals}}} } $$

Solution :
$$\eqalign{ & \sum\limits_{n = 1}^{10} {\sum\limits_{m = 1}^{10} {\left( {{m^2} + {n^2}} \right)} } \cr & = \sum\limits_{n = 1}^{10} {\left[ {\left( {{1^2} + {n^2}} \right) + \left( {{2^2} + {n^2}} \right) + ..... + \left( {{{10}^2} + {n^2}} \right)} \right]} \cr & = 10\left[ {{{\left( 1 \right)}^2} + {{\left( 2 \right)}^2} + ..... + {{\left( {10} \right)}^2}} \right] + 10\left[ {{{\left( 1 \right)}^2} + {{\left( 2 \right)}^2} + ..... + {{\left( {10} \right)}^2}} \right] \cr & = \frac{{20 \cdot 10 \cdot 11 \cdot 21}}{6} = 7700 \cr} $$