Where does the formula from the hypergeometric distribution come from?

Question

In a box are $M$ blue balls and $N$ green balls. We take $n$ balls without putting them back.

Let $X$ be the variable "# blue balls that were taken".

Then $$P\{X = k\} = \frac{\binom{M}{k}\binom{N}{n-k}}{\binom{M+N}{n}}$$

Where does this formula come from?

The textbook then says that $X$ is hypergeometrically distributed.

Answer 1

It is the ratio of the count of ways to select $k$ from $M$ things(eg blue balls) and $n-k$ from $N$ things(eg green balls) to the count of ways to select any $n$ from $M+N$ things(ie all balls); when such selections are made without bias or replacement.

Recollect: $\binom x y$ is the count of ways to select $y$ elements from a set of size $x$, when selecting without replacement.

Answer 2

$\binom{M+N}{n}$ is the number of ways to draw $n$ balls from a collection of $M+N$ (blue or green) balls.

$\binom{M}{k}$ is the number of ways to draw $k$ balls from $M$ blue balls. We drew $n$ balls, so we must have drawn $n-k$ green balls. Hence we multiply this quantity by $\binom{N}{n-k}$ to find the number of ways to draw $k$ blue balls and $n-k$ green balls.