### Video Transcript

Chloe wants to buy eight new books
for her library. There is a choice of 20 fiction
books and 30 nonfiction books. Sheβs going to randomly choose the
eight books, and she wants at least six of them to be nonfiction. Sheβll place the books in the
specific order in which they were selected on a shelf in the library. Which of the following calculations
would lead to the total number of possible orderings of the eight new books? Is it (A) 30π six times 20π two
plus 30π seven times 20π one plus 30π eight? Is it (B) 30πΆ six times 20πΆ two
plus 30πΆ seven times 20πΆ one? Option (C) 30πΆ six times 20πΆ two
plus 30πΆ seven times 20πΆ one plus 30πΆ eight. (D) 30π two plus 20π six plus
30π one plus 20π seven plus 30π eight. Or (E) 30π six times 20π two
times 30π seven times 20π one times 30π eight.

In this question, weβre going to
choose eight books from a total of 20 fiction and 30 nonfiction. Weβre told a limitation on the
number of books which are nonfiction but also that the books are going to be placed
in order. This is a helpful hint of what we
might need to do next.

In order to choose objects from a
collection, we need to think about combinations and permutations. In particular, if we want to choose
π items from a total of π and order does not matter, thatβs a combination. In fact, the calculation we perform
is ππΆπ, sometimes pronounced π choose π, which can be typed into a calculator
or calculated using the formula π factorial over π factorial times π minus π
factorial. If, however, we want to choose π
items from a total of π and order does matter, thatβs a permutation. The calculation we use this time is
πππ, and its formula is just π factorial divided by π minus π factorial.

So, in this question we know the
books are going to be in order, that means we need to use permutations. So, letβs begin with looking at the
ways we could choose the books. Since we want at least six out of
the total of eight to be nonfiction books, that means we could either have six
nonfiction books and two fiction, seven nonfiction books and one fiction, or eight
nonfiction books only.

Starting with the first option, we
know that we need to choose six nonfiction books out of a total of 30. According to our permutation
formula, the number of ways of achieving this when order matters is 30π six. But we also next need to choose the
remaining two fiction books from the group of 20 we have. That must be 20π two.

So then, how do we combine these
two results? Remember, the product rule for
counting, or the basic counting principle, tells us that if there are π ways of
doing something and π ways of doing another thing, then there are π times π ways
of performing both actions. This means there must be 30π six
times 20π two ways of choosing six nonfiction and two fiction books.

Okay, so what about the next
option? Thatβs choosing seven nonfiction
and one fiction book, so that must be 30π seven and 20π one, respectively. Once again, we multiply these to
find the total number of ways of choosing seven nonfiction and one fiction.

Finally, letβs consider the third
choice. Thatβs simply the number of ways of
choosing eight nonfiction from a total of 30, so thatβs 30π eight.

So, we now have the various bits,
but how do we then combine these? The rule of sum is another counting
principle. This tells us that if we have π
ways of doing something and π ways of doing another thing and we cannot do both at
the same time, then there are π plus π ways to choose one of the actions. Hence, we can find the number of
ways of choosing six nonfiction books and two fiction or seven nonfiction books and
one fiction or eight nonfiction books by adding all of our results. That corresponds to option (A),
30π six times 20π two plus 30π seven times 20π one plus 30π eight.