Problem 586

Before Kyne removes hair clips, Ellie has 4 red clips and $4 + 5 + 7 = 16$ clips in total, so the
probability that she randomly chooses a red clip is $\frac{4}{16}$ which equals $\frac{1}{4}$.

After Kyne removes the clips, the probability that Ellie chooses a red
clip is $2 \times \frac{1}{4}$ or
$\frac{1}{2}$.

Since Ellie starts with 4 red clips, then after Kyne removes some clips,
Ellie must have 4, 3, 2, 1, or 0 red clips.

Since the probability that Ellie chooses a red clip is larger than 0,
she cannot have 0 red clips.

Since the probability of her choosing a red clip is $\frac{1}{2}$, then the total number of
clips that she has after $k$ are
removed must be twice the number of red clips, so could be 8, 6, 4, or
2.

Thus, the possible values of $k$ are
$16 - 8 = 8$ or $16 - 6 = 10$ or $16 - 4 = 12$ or $16 - 2 = 14$.

Of these, 12 is one of the given possibilities. (One possibility is that
Kyne removes 2 of the red clips, 5 of the blue clips and 5 of the green
clips, leaving 2 red clips and 2 green clips.)