Coefficient in binomial expansion Whats is the coefficient in front of $x^{50}$ in the binomial expansion $$(1+x)^{1000} + (1 + x)^{999}\cdot x^1 + ... + (1+ x)^1\cdot x^{999} + x^{1000}$$

Hint:
$$\sum_{r=0}^{1000}(1+x)^{1000-r}x^r$$ is a finite geometric series
$$=(1+x)^{1000}\cdot\dfrac{1-\left(\dfrac x{x+1}\right)^{1000+1}}{1-\dfrac x{x+1}}$$
