Question

If ²⁰C_r is the coefficient of x^r in the expansion of (1 + x)²⁰, then the value of `sum_(r = 0)^20 r^2  ""^20C_r` is equal to ______.

Options

• 420 × 2¹⁹

• 420 × 2¹⁸

• 380 × 2¹⁸

• 380 × 2¹⁹

Answer 1

If ²⁰C_r is the coefficient of x^r in the expansion of (1 + x)²⁰, then the value of `sum_(r = 0)^20 r^2  ""^20C_r` is equal to `underlinebb(420 xx 2^18)`.

Explanation:

`sum_(r = 0)^20 r^2  ""^20C_r = sumr(r.""^20C_r) = sum_(r = 1)^20r.20 ""^19C_(r-1)` 

= `20sum_(r = 1)^20(r - 1 + 1)""^19C_(r - 1)`

= `20(sum_(r = 1)^20(r - 1) ""^19C_(r - 1) + sum_(r = 1)^20  ""^19C_(r - 1))`

= `20(19sum_(r = 2)^20 ""^18C_(r - 2) + sum_(r = 1)^20  ""^19C_(r - 1))`

= (380)2¹⁸ + 20.2¹⁹ = 2²⁰(95 + 10)

= (105)2²⁰ = 420 × 2¹⁸