There are 10 boys and six
girls. What is the numerical expression
that allows us to calculate how many ways there are of forming a group that consists
of either three boys or two girls? Is it (A) 10 𝐶 three times six 𝐶
two? Is it (B) 10 𝐶 three plus six 𝐶
two? Is it (C) 10 𝑃 three times six 𝑃
two, (D) 10 𝑃 three plus six 𝑃 two, or (E) 10 𝐶 three minus six 𝐶 two?
We’re forming a group that consists
of either three boys or two girls. So in fact, there are two
events. The first event, let’s call that
event 𝐴, is the event that we choose three boys from a total of 10. The second event, let’s call that
𝐵, is the event that we choose two girls from a total of six. There cannot be a common outcome of
the two events. So they must be mutually
exclusive. This tells us that we’re going to
be able to apply the addition rule to answer this problem.
This says that if two events 𝐴 and
𝐵 are mutually exclusive, where 𝐴 has 𝑚 distinct outcomes and 𝐵 has 𝑛 distinct
outcomes, the total number of outcomes from either of the two events is given by 𝑚
plus 𝑛. We simply add together the numbers
of distinct outcomes from the two events. So our job now is to calculate the
number of outcomes from each event.
We’re choosing three boys from a
total of 10. Now there’s no indication here that
order matters. In fact, let’s say we have boy one,
boy two, and boy three. Switching the order in which we
choose the first two boys so that we choose boy two then boy one then boy three
doesn’t actually matter. We still end up with the same final
three boys. And so we’re dealing with
combinations. The number of combinations there
are, which is when order doesn’t matter, of choosing 𝑟 items from 𝑛 is 𝑛C𝑟 or 𝑛
And the notation that we’ve used
here will not necessarily be the notation you’re used to. Depending on where you are in the
world, you might see both the 𝑛 and the 𝑟 as subscript or alternatively as an
ordered pair and sometimes even as a column vector. So with this in mind, we’ll
calculate the number of ways of choosing three boys from a total of 10. It’s 10 choose three. Similarly, the order in which the
girls are chosen doesn’t matter. And we’re choosing two from a total
of six. So it’s six choose two.
Since the events are mutually
exclusive, we add these together to find the total number of possible outcomes. And so the number of ways of
forming a group consisting of either three boys or two girls is 10 choose three plus
six choose two, which is option (B).