The coefficient of `x^(50)` in the series
`sum_(r=1)^(101)rx^(r-1)(1+x)^(101-r)`is

Options

A. `""^(100)C_(50)`
B. `""^(101)C_(50)`
C. `""^(102)C_(50)`
D. `""^(103)C_(50)`

Answer

b

Explanation

Let `"s" = sum _(r=1)^(101)rx^(r-1)(1+x)^(101-r)`
`= (1+x)^(100) +_ 2 x (1+x)^(99) + 3 x ^(2) (1+x)^(98) +... + 101 x^(100)`
`"S=(1+x)^(100){1+2(x/(1+x))+3(x/(1+x))+...+101(x/(1+x))^(100)}` …(i)
`therefore (Sx)/(1+x)= (1+x)^(100){{:((x/(1+x))+2(x/(1+x)_)^(2)),(+3(x/(1+x))^(3)+...+101(x/(1+x))^(101)):}}` ...(ii)
On subtracting Eq. (ii) from Eq. (i) then we get
` (Sx)/(1+x)= (1+x)^(100){{:(1+(x/(1+x))+(x/(1+x))^(2)),(+...+(x/(1+x))^(3)-101(x/(1+x))^(101)):}}`
`=(1+x)^(100){(1cdot(1-(x/(1+x))^(101)))/(1-(x/(1+x)))-101 (x/(1+x))^(101)}`
`therefore S= (1 + x)^(101) - x ^(101) (1+x)-101x^(101)`
and coefficient of `x^(50) " in S" = ""^(101)C_(50)`.