The value of ${\left({ }^{15}{\mathrm{C}}_0^{ }\right)}^2-{\left({ }^{15}{\mathrm{C}}_1^{ }\right)}^2+{\left({ }^{15}{\mathrm{C}}_2^{ }\right)}^2-\cdots -{\left({ }^{15}{\mathrm{C}}_{15}^{ }\right)}^2$ is

As we know that ${{\mathrm{C}}_0}^2{-}^{ }{\mathrm{C}}_1^2+{\mathrm{C}}_2^2{-}^{ }{\mathrm{C}}_3^2+\cdots +(-1{)}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}^2=0$

(if n is odd) and in the question n = 15 (odd). 
Hence, sum of given series is 0.