The correct answer is (b) \(\Sigma_{r = 0}^{r = 1000}\)(100Cr x^1000 – r y^r)
To elaborate: The expansion can be done using binomial theorem.
(x + y)^1000 = 1000C0 x^1000 y^0 + 1000C1 x^999 y^1 +….+ 1000C999 x^1 y^999 + 1000C1000 x^0 y^1000
This can also be written as,
(x + y)^1000 = \(\Sigma_{r = 0}^{r = 1000}\)(100Cr x^1000 – r y^r).