The sum of the coefficients of all the integral powers of x in the expansion of (1+2x)80 is

The coefficients of the integral powers of x are 80C0,80C2⋅22,80C4⋅24,…,80C80⋅280Now, (1+2)80=80C0+80C1⋅2+80C2⋅22+…+80C80⋅280           ……. (1)and (1−2)80=80C0−80C12+80C2⋅22−…+80C80⋅280               …….. (2)Adding Eq. (1) and (2), we get380+1=2 80C0+80C2⋅22+80C4⋅24+…+80C80⋅280∴ 80C0+80C2⋅22+80C4⋅24+…+80C80⋅280=12380+1