$4$ chips are selected at random from a box of $30$ chips

Question

4 chips are selected at random and without replacement from a box containing 5 black, 5 white, 5 red, 5 yellow, 5 blue and 5 green chips. Find the probabilities of the following events

(a) Two of the chips are white, while the other two are of a second color

(b) The selected chips are of three distinct colors.

I am having a lot of issues with problems like this. I don't know how I should think to solve them!

For $(a)$, I thought the following way:

There are $6$ colors to choose from for two balls, so we have $6\choose 1$ Having chosen a color, we need two balls from that color so there are $5\choose 2$ ways to pick $2$ balls of that color. Then there are $5$ colors to choose from for the second ball and third ball, so $5\choose 2$ ways of picking two colors of the remaining $5$ colors. Having picked two colors (other than white), there are $5\choose 1$ ways to pick one ball from one of those two colors, and $5\choose 1$ ways to pick one ball from the other of those two colors. And now I feel like I have answered question $(b)$ and not $(a)$.

I am extremely confused, and any help would be immensely appreciated!!

Answer 1

4 chips are selected at random and without replacement from a box containing 5 black, 5 white, 5 red, 5 yellow, 5 blue and 5 green chips. Find the probability that two of the chips are white, and the other two are of a second color.

There are $\binom{30}{4}$ ways to select four of the thirty chips.

For the favorable cases, choose two of the five white chips, one of the other five colors, and two of the five chips of that color, which can be done in $$\binom{5}{2}\binom{5}{1}\binom{5}{2}$$ ways.

Hence, the probability that two of the four selected chips are white, and the other two are of a second color is $$\frac{\dbinom{5}{2}\dbinom{5}{1}\dbinom{5}{2}}{\dbinom{30}{4}}$$

4 chips are selected at random and without replacement from a box containing 5 black, 5 white, 5 red, 5 yellow, 5 blue and 5 green chips. Find the probability that the selected chips are of three different colors.

Notice that this can only occur if we select two chips of one color and one chip each from two other colors.

Select the color from which two chips are drawn, select two chips of that color, select two colors from which one chip will be drawn (the colors must be different from the color from which two chips are drawn), and select one chip from each of those colors.

This can be done in $$\binom{6}{1}\binom{5}{2}\binom{5}{2}\binom{5}{1}^2$$ ways.

Divide by the number of ways of selecting four of the $30$ available chips to get the probability that the four chips are drawn from three different colors.