A bag contains 3 red marbles 3 green ones 1 lavender one 3 y

A bag contains 3 red marbles, 3 green ones, 1 lavender one, 3 yellows, and 2 orange marbles.

How many sets of five marbles include at least two red ones?

Solution

Whenever you see the words “at least” or “at most,” you are probably going to have to consider more than one case. Here, we have to consider two separate cases: case 1 is the case where we get two red ones, and case 2 is the case where we get three red marbles (“at least” means we get that number or more than that number). For case 1, we want two of the three red marbles, and there are 3C2 = 3 ways to do this

Then, we still need to get three more marbles to make a set of five. The remaining three can’t be red, so we must choose them all from the non-red marbles. There are nine non-red marbles, so there are 9C3 = 84 ways to do this. The total number of ways to choose a set of five marbles where two of them are red is

3C2 * 9C3 = 3*84 = 252

For case 2, there is 3C3 = 1 way to get all three red marbles. Then, we still need two more to get our set of five. Again, these last two can’t be red, so we must choose them from the nine nonred ones. There are 9C2 = 36

ways to do this. So, there are

3C3 * 9C2 = 1*36 = 36 ways to choose a set of five marbles where three of them are red.

These two cases are disjoint (meaning they can’t both happen at the same time – you can’t have a set of five marbles where two are red and three are red at the same time). To find the total number of ways to get at least two red, we use the addition principle and add the results of the two cases together. This gives 36 +252 = 288 ways to do this.