Question Video: Counting Outcomes of Two Events Using the Addition Rule Mathematics

There are 10 boys and 6 girls. What is the numerical expression that allows us to calculate how many ways there are of forming a group that consists of either 3 boys or 2 girls?

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Video Transcript

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 choose 𝑟.

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).

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