Video Transcript
How many ways can two pencils of the same color be selected from six red and three
blue?
In order to answer this question, let’s think about what we know about each color
pencil. We have six red and three blue pencils. And so the event choosing a red pencil is mutually exclusive with the event choosing
a blue pencil. In other words, if we choose a pencil, it cannot be simultaneously red and blue at
the same time. And this is really useful, because when we’re working with mutually exclusive events,
we can use the addition rule. Now this tells us that if A and B are mutually exclusive events, where event A has 𝑚
distinct outcomes and event B has 𝑛 distinct outcomes, then there are 𝑚 plus 𝑛
distinct outcomes from either A or B.
And so let’s begin by identifying the possible events. We’re choosing pencils of the same color. So event A might be the event of choosing two red pencils. Then, event B will be the event choosing two blue pencils. We can see that if we can identify the number of outcomes of choosing two red pencils
and the number of outcomes of choosing two blue pencils, then we’ll simply add them
together to find the total number of combinations.
We’re choosing two pencils from a total of six. And of course here order doesn’t matter. So we’re interested in combinations. Specifically, there are six choose two ways of choosing our two red pencils. In a similar way, the order in which we choose the blue pencils does not matter. So there are three choose two ways of choosing two blue pencils.
The addition rule tells us that the total number of ways of choosing two pencils of
the same color is found by adding these two numbers, in other words six choose two
plus three choose two. And so there are six choose two plus three choose two ways to choose two pencils of
the same color from six red and three blue.
