If a fair die is rolled $1000$ times, what is the probability that a six is obtained between $150$ and $167$ times, what is the probability of getting 200 sixes?
My logic: Noticing the probability of rolling $2$ $6$'s out of $3$ rolls was ${3\choose 2}*{\frac{1}{6}}^2 *\frac{5}{6}$ and other similar examples, I assumed they all took form $ Pr($Rolling A 6's out of B rolls)${B\choose A}*{(\frac{1}{6}})^A *(\frac{5}{6})^{B-A}$. So I could take a Riemann sum of that from $151$ to $166$: $$\sum_{i = 151}^{166} {1000\choose i}*{(\frac{1}{6}})^i *(\frac{5}{6})^{1000-i} $$ Which yielded result $\approx .41443$ which seems roughly right.
Next I just want that probability of getting $200$ sixes $${1000\choose 200}*{(\frac{1}{6}})^{200} *(\frac{5}{6})^{800} = .0007$$
Is this right? Could this have been calculated more easily?