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Video: Using Combinations to Determine the Probability of an Event

Bethani Gasparine

Marilyn invited 4 of her 24 friends over for a movie night. If she chose to invite her friends at random, what is the probability that Cynthia, Ashley, Angelina, and Sandra are chosen?

02:03

Video Transcript

Marilyn invited four of her 24 friends over for a movie night. If she chose to invite her friends at random, what is the probability that Cynthia, Ashley, Angelina, and Sandra are chosen?

The number of ways we can choose four friends from 24 friends is by 24 choose four. The formula for combination is 𝑛 choose π‘Ÿ, where 𝑛 in this case will be 24 friends and we’re choosing four of them, that would be π‘Ÿ, which is equal to 𝑛 factorial divided by π‘Ÿ factorial times 𝑛 minus π‘Ÿ factorial, which means we have 24 factorial divided by four factorial times 24 minus four factorial.

So we have 24 factorial divided by four factorial times 20 factorial. Twenty-four factorial is 24 times 23 times 22 times 21 all the way down to one. And then four factorial is four times three times two times one. And 20 factorial is 20 times 19 times 18 times 17 all the way down to one.

Here we can see the twenties, the nineteens, the eighteens, the seventeens, all the way down to ones can cancel. So we have 24 times 23 times 22 times 21, which is 255024 all over four times three times two times one, which is 24. And now we divide these numbers and we get 10626. So the probability of these girls being chosen, where the order doesn’t matter, would be one out of 10626.