Let 1+x=r
∴S=1⋅r+2⋅r2+3⋅r3+…+100r100 ......(1) (AGP)
rS=1⋅r2+2⋅r3+…+99r100+100r101 ......(2)
(1) − (2) gives
S=−x2(1+x)101+x21+x100(1+x)101
∴ coefficient x48 in S
=− coefficient of x48 in x2(1+x)101+100⋅ Coefficient of x48 in x(1+x)101
=100⋅101C49−101C50