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

What is the numerical expression we would use to find in how many ways can 4 balls of the same color be selected from 10 blue balls, 6 green balls, and 7 red balls? Assume none of the balls are identical.

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### Video Transcript

What is the numerical expression we would use to find in how many ways can four balls of the same color be selected from 10 blue balls, six green balls, and seven red balls. Assume none of the balls are identical. (A) 10 𝐶 four times six 𝐶 four times seven 𝐶 four, (B) 10 𝑃 four times six 𝑃 four times seven 𝑃 four. Is it (C) 10 𝑃 four plus six 𝑃 four plus seven 𝑃 four, (D) 10 𝐶 four plus six 𝐶 four plus seven 𝐶 four, or (E) 10 𝐶 four times six 𝐶 four plus seven 𝐶 four?

We’re selecting four balls from 10 blue, six green, and seven red. Now the key aspect of this question, which will help us answer it, is that none of the balls are going to be identical. So when we choose four balls, we’re either going to choose four blue, four green, or four red. Since no outcome is shared by the event of choosing four blue balls, four green balls, and so on, the three events are said to be pairwise mutually exclusive.

The addition rule says that the number of distinct outcomes from this collection of pairwise mutually exclusive events is the sum of the number of distinct outcomes from each event. So we need to work out the number of ways of choosing four blue balls from a total of 10, four green balls from a total of six, and four red balls from a total of seven. And then we’ll add these values together.

Now since the order in which these balls are chosen doesn’t matter — choosing, say, four blue balls will result in the same final outcome regardless of the order in which these are chosen — we know that we have combinations. In other words, the number of ways of choosing 𝑟 items from a total of 𝑛 distinct items when order doesn’t matter is 𝑛 choose 𝑟. So the number of ways of choosing the four blue balls from a total of 10 is 10 choose four. The number of ways of choosing four green balls from a total of six is six choose four. And choosing four red balls from a total of seven is seven choose four.

The addition rule says that the total number of outcomes is the sum of these. So the numerical expression we will use is 10 𝐶 four plus six 𝐶 four plus seven 𝐶 four. And that’s option (D).

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