The number of terms which are free from radical signs in the expansion of (x^1/5+y^1/10)⁵⁵ is

If the third term in the expansion of (1/x+ xlog₁₀ x)⁵ is 1, then x=

2.C₀+2².C₁/2+2³.C₃/3+……..+2ⁿ⁺¹.C_n/n+1 =

If n is odd then C₀²-C₁²+C₂²-……….+(-1)ⁿ C_n² =

Coefficient of x⁵⁰ in (1+x)¹⁰⁰⁰+2x(1+x)⁹⁹⁹+3x²(1+x)⁹⁹⁸+……..+1001x¹⁰⁰⁰ is