The sum of the products of the ten numbers $$ \pm 1, \pm 2, \pm 3, \pm 4, \pm 5$$     taking two at a time is

Solution :
$${\left( {1 - 1 + 2 - 2 + ..... + 5 - 5} \right)^2} = {1^2} + {1^2} + {2^2} + {2^2} + ..... + {5^2} + {5^2} + 2S,$$              where $$S$$ is the required sum
or, $$0 = 2\left( {{1^2} + {2^2} + {3^2} + {4^2} + {5^2}} \right) + 2S.$$