Video Transcript
In a final exam that consists of 12
questions, a quarter of them are essay questions and the rest are multiple-choice
questions. A student has to solve 10 of the
questions, where at least seven of them are multiple-choice questions and the rest
are essay questions. Write the calculation that would
give the number of ways that the student can choose which questions to answer. Is it (A) 10 choose seven times
three choose three times 10 choose eight times three choose two times 10 choose nine
times three choose one? Is it (B) nine choose seven plus
three choose three plus nine choose eight plus three choose two plus nine choose
nine plus three choose one? (C) 12 choose seven times three
choose three plus 12 choose eight times three choose two plus 12 choose nine times
three choose one. (D) is nine choose seven times
three choose three plus nine choose eight times three choose two plus nine choose
nine times three choose one. Or finally, is it option (E) nine
choose seven plus 12 choose three plus nine choose eight times 12 choose two plus
nine choose nine plus 12 choose one?
Let’s begin by working out what the
required outcome here is. In order to complete the final
exam, a student has to solve 10 questions. At least seven of these 10
questions must be multiple choice, whilst the rest are essay questions. We’re also told that there are 12
to choose from. So let’s begin by working out how
many of these are essay and how many are multiple choice. In fact, we’re told that a quarter
of the questions are essay questions. A quarter of 12 is three, so out of
the 12 questions in total, three of those are essay questions. The rest of these are multiple
choice.
So we can calculate this in one of
two ways. We could subtract three from 12 or
alternatively find three-quarters of 12. Either way, we find that nine of
the questions are multiple choice. So with this in mind, let’s find
the possible outcomes that would lead a student to solving 10 questions. They could, of course, solve
exactly seven multiple-choice questions. And so the remaining three, to make
up 10, would need to be essay questions. Alternatively, they could solve
eight multiple-choice questions and just two essay questions. Finally, they could choose nine
multiple-choice and one essay question.
It might be tempting to think that
there’s a fourth option, 10 multiple choice and no essays. But remember, we said that there
are a total of nine multiple-choice questions, so that’s the absolute maximum number
of multiple-choice questions a student could answer.
So with this in mind, how do we
calculate the number of ways of choosing, for instance, seven multiple-choice and
three essay questions? Let’s first look at the individual
events in each option. We can choose multiple-choice
questions, and we can choose essay questions. These, of course, are independent
events. Choosing, for instance, a
multiple-choice question doesn’t actually change the number of essay questions we
have to choose from. This means in order to find the
number of ways of choosing seven multiple-choice questions and three essay
questions, we can use the fundamental counting principle.
This tells us that if we have two
independent events 𝐴 and 𝐵, where the number of possible outcomes for 𝐴 is 𝑚 and
the number of outcomes for 𝐵 is 𝑛, the total number of possible outcomes for the
two events together is 𝑚 times 𝑛. So if we can find the number of
ways of choosing seven multiple-choice questions, we can multiply that by the number
of ways of choosing three essay questions. That will give us the total number
of possible ways of choosing seven multiple-choice and three essay questions.
Now, specifically, we’re looking to
choose seven multiple-choice questions from the total of nine. Now, remember, the number of ways
to select 𝑟 objects from a total of 𝑛 when order doesn’t matter is 𝑛 choose
𝑟. Now here order doesn’t matter, so
the number of ways of choosing seven multiple-choice questions from a total of nine
is nine choose seven.
In a similar way, we’re looking to
choose three essay questions from a total of three. So there are three choose three
ways of doing this. The fundamental counting principle,
sometimes called the product rule for counting, tells us that the total number of
outcomes then is nine choose seven times three choose three. Let’s repeat this process for
option two, the number of ways of choosing eight multiple-choice questions and two
essay questions. The number of ways of choosing
eight multiple-choice questions will be nine choose eight. And the number of ways of choosing
the essay questions is three choose two. So the total number of outcomes for
option two is nine choose eight times three choose two.
We can now calculate the number of
ways of choosing nine multiple-choice questions and one essay question. This time, that’s nine choose nine
times three choose one. So we have the number of ways of
choosing seven multiple choice and three essay, eight multiple choice and two essay,
and nine multiple choice and one essay. So how do we combine these and work
out the number of ways that the student can choose which questions to answer?
Well, first, it’s worth noting that
these options are pairwise mutually exclusive. In other words, a student cannot
simultaneously choose seven multiple-choice questions and three essay questions as
well as choosing eight multiple-choice questions and two essay questions. And when we’re working with
pairwise mutually exclusive events, we can find the total number of outcomes by
using the addition rule.
Let’s say once again that 𝐴 and 𝐵
are two events. This time, they’re pairwise
mutually exclusive. If 𝐴 has 𝑚 outcomes and 𝐵 has
𝑛, the number of outcomes from 𝐴 or 𝐵 is 𝑚 plus 𝑛. And that’s really useful because
this tells us the total number of ways that the student can choose which questions
to answer is found by summing the three results we saw earlier, so nine choose seven
times three choose three plus nine choose eight times three choose two plus nine
choose nine times three choose one.
And if we look through our
multiple-choice options, that’s option (D). This is the calculation that gives
the number of ways that the student can choose which questions to answer.
