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).
