Official solution
\[{\left( {1 - 2\sqrt x } \right)^{50}}\]
= \[{}^{50}{C_0} + {}^{50}{C_1}.\left( { - 2\sqrt x } \right) + {}^{50}{C_2}.{\left( { - 2\sqrt x } \right)^2} + ....\]
Now we need to find out those coefficient where degree of x is integer and you can see at odd terms power of x is integer.
Let \[{\left( {1 - 2\sqrt x } \right)^{50}}\] = Odd(A) - Even(B)
So \[{\left( {1 + 2\sqrt x } \right)^{50}}\] = A + B
\[\therefore\] 2A = \[{\left( {1 + 2\sqrt x } \right)^{50}}\] + \[{\left( {1 - 2\sqrt x } \right)^{50}}\]
\[ \Rightarrow A = {1 \over 2}\left[ {{{\left( {1 + 2\sqrt x } \right)}^{50}} + {{\left( {1 - 2\sqrt x } \right)}^{50}}} \right]\]
Now to find sum of coefficient of A, put x = 1.
\[\therefore\] Sum of coefficient of A = \[{1 \over 2}\left[ {{{\left( {1 + 2} \right)}^{50}} + {{\left( {1 - 2} \right)}^{50}}} \right]\]
= \[{1 \over 2}\left[ {{{\left( 3 \right)}^{50}} + 1} \right]\]