Video Transcript
A cup contains 10 blue marbles, six
green marbles, and seven red marbles. None of the marbles in the cup are
identical. How many ways can four marbles be
chosen from the cup so that exactly three of them are the same color?
Let’s begin by identifying the
different events that will lead to an outcome of four marbles where exactly three of
them are the same. We can select four marbles so
exactly three of them are blue. We can select four where exactly
three of them are green or four where exactly three of them are red. Now we notice that no outcomes are
shared by the different events. So we say that these events are
pairwise mutually exclusive.
In this case, we can use the
addition rule. This says that the number of
distinct outcomes from a collection of pairwise mutually exclusive events is the sum
of the number of distinct outcomes from each event. So we need to calculate the number
of outcomes in each event. Let’s start with the first, that’s
choosing three blue marbles. Essentially, in this case, we’re
choosing three blue marbles and one that is not blue. Now, in fact, these two events are
independent; a specific outcome of one doesn’t affect the number of possible
outcomes of the other. And so we can use the fundamental
counting principle. That tells us that the number of
outcomes of the two events together is found by multiplying the number of outcomes
from each event.
There are 10 blue marbles, and
there are six plus seven marbles that are not blue. That’s 13. This means that there are 13 ways
of choosing one marble that is not blue. It’s a little bit more complicated
when it comes to choosing three marbles from a total of 10. The order in which we choose these
marbles doesn’t matter. And so we say that there are 10 𝐶
or 10 choose three ways of choosing three marbles from a total of 10. The fundamental counting principle
tells us then the number of ways to select four marbles so that exactly three of
them are blue is 13 times 10 choose three.
We will now move on to the green
marbles. There are six green marbles
altogether and then 17 which are not green. So there are six choose three ways
of choosing three green marbles from a total of six. And then by the fundamental
counting principle, the number of ways of selecting four so that exactly three of
them are green is 17 times six choose three.
Finally, we have the red
marbles. Seven are red, and 16 are not
red. That means there are seven choose
three ways of choosing those three red marbles from a total of seven. There are 16 ways of choosing a
marble that’s not red. And so the fundamental counting
principle says that the number of ways to select four marbles so that exactly three
of them are red is 16 times seven choose three.
Finally, we apply the addition
rule. So the number of different ways
that four marbles can be chosen from the cup so that exactly three of them are the
same color is the sum of these. It’s 13 times 10 choose three plus
17 times six choose three plus 16 times seven choose three. That’s option (C).
