WEBVTT
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A class contains 27 boys and 29 girls.
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In how many ways can you select a team of four people from the class such that every member of the team is of the same sex?
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We’ll need to know how many ways we can get a group of four boys, and we’ll add that value to the number of ways we’ll get a group of four girls.
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In a four-person team, order doesn’t matter, and that means this is a combination.
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The formula for this is a combination of 𝑛 objects, where we’re choosing 𝑟, number of them, equals 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial.
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The combination of boys is 27 boys, where we’re choosing four, four factorial times 27 minus four factorial.
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27 minus four equals 23.
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We need to follow this same setup for the girls.
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We’re looking for the combination of 29 girls, taking four at a time.
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It’ll be 29 factorial over four factorial times 29 minus four factorial.
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29 minus four is 25.
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If we bring down our equations, we can start to simplify them.
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Remember what factorial means.
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27 factorial is equal to 27 times 26 times 25 times 24, all the way down to one.
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We could also say that 27 factorial is equal to 27 times 26 times 25 times 24 times 23 factorial.
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The reason we write it like this is to notice that something in the numerator and the denominator cancels out.
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There’s a 23 factorial in the numerator and the denominator.
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After those cancel out, we’re left with 27 times 26 times 25 times 24 over four factorial, which can be rewritten as four times three times two times one.
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We’ll do some more simplifying.
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24 divided by two equals 12, 27 divided by three equals nine, and 12 divided by four equals three.
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To find the combinations, we’ll have to multiply nine times 26 times 25 times three.
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It equals 17550.
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Now we wanna simplify the combination of girls.
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We’ll take our 29 factorial and rewrite it as 29 times 28 times 27 times 26 times 25 factorial.
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And then the 25 factorial in the numerator and the denominator cancel out.
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We need to expand the four factorial: four times three times two times one.
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We can then simplify.
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28 divided by four equals seven, 27 divided by three equals nine, and 26 divided by two equals 13.
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To find the combination of girls, multiply 29 times seven times nine times 13, which equals 23751.
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To find the total way we could select a team of four people from the class, we need to add the boy teams and the girl teams.
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Adding them together, we get 41301.
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There are 41301 ways we can select a team of four people from the class such that every member of the team is of the same sex.