A bag contains 3 red marbles, 2 green ones, 1 lavender one, 2
yellows, and 2 orange marbles.
a) How many possible sets of four marbles are there?
Solution: The question can be reworded as “how many
ways can you choose four marbles from a set of ten
marbles?” There are
ways to do this.
b) How many sets of three marbles include all the red ones?
Solution: To include all the red ones, we ask “how many
ways can you choose three marbles from a set of three?”
There is
way to do this.
c) How many sets of four marbles include none of the red
ones?
Solution: If you don’t include any of the red ones, you are
choosing all four marbles from the non-red marbles. Since
there are seven non-red marbles, there are
ways to do this.
d) How many sets of four marbles include one of each color
other than lavender?
Solution: You want to choose one of each color besides
lavender. There are
ways to choose one of the
red ones,
ways to choose one green one, and
the same number of ways to choose one yellow and one
orange (since there are two of each of those colors). Using
the multiplication principle, there are
ways to choose a set of four marbles that include one of
each color besides lavender.
e) 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
(disjoint) 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
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
seven non-red marbles, so there are
ways to
do this. The total number of ways to choose a set of five
marbles where two of them are red is
. For case 2, there is
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 seven nonred ones. There are
ways to do this. So, there
are
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
ways to do this.
f) How many sets of five marbles include either the lavender
one or exactly one yellow but not both colors?
Solution: Again, we have to consider two cases here. Case
one is where we get the lavender marble, and case two is
where we get exactly one yellow marble. They are treated
separately because it says “but not both colors,” so these
two cases can’t happen at the same time (so they are
disjoint). For case 1, we need to get the one lavender
marble. There is
way to do this. The other four
marbles must come from the marbles that aren’t lavender
and aren’t yellow. There are seven such marbles, so there
are
ways to choose the last four marbles. In
total, there are
ways to
choose five marbles where one of them is the lavender
one and none are yellow. For case 2, we need to get
exactly one yellow marble. There are
ways to
do this. Again, the other four marbles can’t be yellow and
can’t be lavender, so there are
ways to
choose the last four marbles. In total, there are
ways to get exactly one
yellow marble. Again, we use the addition principle to find
the total number of ways to choose a set of five marbles
that include either the lavender one or exactly one yellow
one, but not both. This gives
ways to do
this.