The value of $C_1 ^2+C_2 ^2....+C_n ^2$ (where $C_i$ is the $i^{th}$ coefficient of $(1+x)^n$ expansion), is:

${ \left( 1+x \right) }^{ n }=^{ n }{ { C }_{ 0 } }+x\left( ^{ n }{ { C }_{ 1 } } \right) { +{ x }^{ 2 }\left( ^{ n }{ { C }_{ 2 } } \right) + }{ x }^{ 3 }\left( ^{ n }{ { C }_{ 3 } } \right) +...+{ x }^{ n }\left( ^{ n }{ { C }_{ n } } \right) \\$

${ \left( x+1 \right) }^{ n }={ x }^{ n }\left( ^{ n }{ { C }_{ 0 } } \right) +{ x }^{ n-1 }\left( ^{ n }{ { C }_{ 1 } } \right) +...+{ x }^{ 0 }\left( ^{ n }{ { C }_{ n } } \right)$

Now, $\quad \sum _{ r=0 }^{ n }{ { \left( ^{ n }{ { C }_{ r } } \right) }^{ 2 } }$= co-efficient of $x^n$ in $(1+x)^{2n}=^{ 2n }{ { C }_{ n } }$

Thus, $^{ 2n }{ { C }_{ n } }={ \left( ^{ n }{ { C }_{ 0 } } \right) }^{ 2 }+{ \left( ^{ n }{ { C }_{ 1 } } \right) }^{ 2 }+...{ \left( ^{ n }{ { C }_{ n } } \right) }^{ 2 }$, where ${ \left( ^{ n }{ { C }_{ i } } \right) }$ is the $i^{th}$ coefficient of $(1+x)^n$ expansion.

So, $C_1^2+C_2^2+...+C_n^2={ \left( ^{ 2n }{ { C }_{ n } } \right) }=\dfrac { 2n! }{ n!n! }$