Computing probability of extracting AT LEAST n balls from a urn

Question

I cannot figure out how to model the probability computation in the following scenario.

I have an urn with 31 balls, of which 21 are red and 10 are black. If I extract 10 balls from the urn, what is the probability that AT LEAST 6 are black?

(edit: it's a bulk extraction, so no replacement or order matter)

I am more interested in how to use binomial formulas than in the numeric result; thanks!

Answer 1

Preassuming that this is done without replacement:$$\sum_{n=6}^{10}p_n$$where: $$p_n=\frac{\binom{10}n\binom{21}{10-n}}{\binom{31}{10}}$$is the probability that exactly $n$ are black.