You have $4$ possibilities for the drawings, RRR, RRB, RBR and BRR.
Calculate each probability separately:
$$RRR: \frac{n-k}{n}\cdot\frac{n-k-1}{n-1}\cdot\frac{n-k-2}{n-2}$$
$$RRB: \frac{n-k}{n}\cdot\frac{n-k-1}{n-1}\cdot\frac{k}{n-2}$$
$$RBR: \frac{n-k}{n}\cdot\frac{k}{n-1}\cdot\frac{n-k-1}{n-2}$$
$$BRR: \frac{k}{n}\cdot\frac{n-k}{n-1}\cdot\frac{n-k-1}{n-2}$$
The denominator in each case is the same, $\dfrac1{n(n-1)(n-2)}$, and we can see that each numerator contains a common factor of $(n-k)(n-k-1)$, so that we can write the sum of these probabilities as:
$$\dfrac{(n-k)(n-k-1)}{n(n-1)(n-2)}(n-k-2+3k)=\dfrac{(n-k)(n-k-1)(n+2k-2)}{n(n-1)(n-2)}$$
which is the probability that you want.
We can prove this by considering the probabilities that we don't get at least two reds, and sum them. This gives:
$$\dfrac{k(k-1)}{n(n-1)(n-2)}(3(n-k)+k-2)=\dfrac{k(k-1)(3n-2k-2)}{n(n-1)(n-2)}$$
If we add the two probabilities together, we get $1$.
These events themselves are not independent, as $P(RRR\cap RRB)\ne P(RRR)P(RRB)$, but whether we pull a red or black ball next is only dependent on the proportion of red to black balls left.