Discreate mathematics

Question

Is there any one who can help me past of days or weeks 🥺 I didn't get answer from here plz respond anyone for probability questions :

A box contains 17 balls of each color (i.e., green, red, and blue). If 5 balls are to be drawn randomly

From the box, what is the probability of having 3 balls of the same color?

Answer 1

If X is a set containing t elements, the number of unordered, k-element selections from X, allowing repetitions, is C(k + t - 1, t - 1).

For this example, first find the size of the sample space. There are C(5 + 3 - 1, 3 - 1) = C(7, 2) = 21 ways of selecting 5 balls.

There are two possible questions: draw exactly 3 balls of the same color or at least 3 of the same color. Drawing at least three balls of the same color is an easier question. How could you not draw 3 balls of the same color? You would have to have 2G2R1B, 2G1R2B, or 1G2R2B, so then the probability of at least 3 balls of the same color would be 1 - 3/21 = 18/21.

If the question was to find how many ways you can draw 5 balls with exactly 3 of the same color, suppose you want exactly 3 green balls. Then you are choosing 2 balls from two colors and there are C(2 + 2 - 1, 2 - 1) = C(3, 1) = 3 ways of doing this. Explicitly the combinations are: 3G2R, 3G1R1B, 3G2B. In a similar way, there are 3 ways of having 3 red balls and 3 ways of having 3 blue balls, so the probability is 9/21 = 3/7.

The number 17 is not significant unless the balls are numbered, in which case this is a completely different problem.