Video Transcript
Chloe wants to buy eight new books
for her library. There is a choice of 20 fiction
books and 30 nonfiction books. She’s going to randomly choose the
eight books, and she wants at least six of them to be nonfiction. She’ll 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) 30𝑃 six times 20𝑃 two
plus 30𝑃 seven times 20𝑃 one plus 30𝑃 eight? Is it (B) 30𝐶 six times 20𝐶 two
plus 30𝐶 seven times 20𝐶 one? Option (C) 30𝐶 six times 20𝐶 two
plus 30𝐶 seven times 20𝐶 one plus 30𝐶 eight. (D) 30𝑃 two plus 20𝑃 six plus
30𝑃 one plus 20𝑃 seven plus 30𝑃 eight. Or (E) 30𝑃 six times 20𝑃 two
times 30𝑃 seven times 20𝑃 one times 30𝑃 eight.
In this question, we’re going to
choose eight books from a total of 20 fiction and 30 nonfiction. We’re 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
𝑟 items from a total of 𝑛 and order does not matter, that’s a combination. In fact, the calculation we perform
is 𝑛𝐶𝑟, sometimes pronounced 𝑛 choose 𝑟, which can be typed into a calculator
or calculated using the formula 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟
factorial. If, however, we want to choose 𝑟
items from a total of 𝑛 and order does matter, that’s a permutation. The calculation we use this time is
𝑛𝑃𝑟, and its formula is just 𝑛 factorial divided by 𝑛 minus 𝑟 factorial.
So, in this question we know the
books are going to be in order, that means we need to use permutations. So, let’s 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 30𝑃 six. But we also next need to choose the
remaining two fiction books from the group of 20 we have. That must be 20𝑃 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 𝑎 ways of
doing something and 𝑏 ways of doing another thing, then there are 𝑎 times 𝑏 ways
of performing both actions. This means there must be 30𝑃 six
times 20𝑃 two ways of choosing six nonfiction and two fiction books.
Okay, so what about the next
option? That’s choosing seven nonfiction
and one fiction book, so that must be 30𝑃 seven and 20𝑃 one, respectively. Once again, we multiply these to
find the total number of ways of choosing seven nonfiction and one fiction.
Finally, let’s consider the third
choice. That’s simply the number of ways of
choosing eight nonfiction from a total of 30, so that’s 30𝑃 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 𝑎
ways of doing something and 𝑏 ways of doing another thing and we cannot do both at
the same time, then there are 𝑎 plus 𝑏 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),
30𝑃 six times 20𝑃 two plus 30𝑃 seven times 20𝑃 one plus 30𝑃 eight.
