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In how many different ways can 12 people be picked from 15?
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In this question, weโre looking to choose 12 people from a total of 15.
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Thereโs no indication that the order in which we choose these 12 people matters.
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In mathematics, we call this a combination.
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Itโs a way to calculate the total outcomes of an event where the order of the outcomes does not matter.
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And if weโre looking to find the number of ways of choosing our items from a total of ๐ items, we use ๐ choose ๐, where ๐ choose ๐ is ๐ factorial over ๐ factorial times ๐ minus ๐ factorial.
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And ๐ factorial is ๐ times ๐ minus one times ๐ minus two times ๐ minus three, and so on.
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Now, weโre looking in this question to choose 12 people from a total of 15.
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So, in our formula, weโre going to let ๐ be equal to 12 and ๐ be equal to 15.
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And so, the calculation we need to perform is 15 choose 12.
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Thatโs 15 factorial over 12 factorial times 15 minus 12 factorial.
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But of course, 15 minus 12 is three.
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So, we have 15 factorial over 12 factorial times three factorial.
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Now, generally, when weโre calculating combinations, we want to try and avoid evaluating our factorials fully.
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So, we wouldnโt really want to work out 15 times 14 times 13 times 12, and so on.
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Instead, we spot that 15 factorial can be written as 15 times 14 times 13 times 12 factorial.
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And then, we see that we can divide both our numerator and denominator by 12 factorial.
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We need to look for some further common factors.
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Well, we can divide both 14 and two by two and 15 and three by three.
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And so, 15 choose 12 simplifies to five times seven times 13 divided by one or just five times seven times 13.
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This is 455.
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And so, by calculating 15 choose 12, weโve worked out the number of ways that 12 people can be picked from 15.
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Itโs 455.