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Find the value of π given that πP four equals 24.
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We know that πPπ equals π factorial over π minus π factorial and that πP four equals 24, which means we donβt know how many items our original set held, but we do know weβre selecting four of them.
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In cases where we arenβt given the π-value, it might not seem obvious where we should start, so letβs start by plugging in what we know into our formula, which will give us πP four equals π factorial over π minus four factorial.
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We know that π factorial equals π times π minus one factorial.
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We can use this property to rewrite our numerator so that π factorial is equal to π times π minus one factorial.
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But that doesnβt get us any closer to simplifying the fraction.
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But we can expand π minus one factorial, which would be π minus one times π minus one minus one factorial, which would be π minus two factorial.
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And we could expand π minus two factorial to be π minus two times π minus three factorial.
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And π minus three factorial would be equal to π minus three times π minus four factorial.
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This allows us to cancel out the π minus four factorial in the numerator and the denominator.
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And since we know that πP four equals 24, we can say that 24 will be equal to π times π minus one times π minus two times π minus three.
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This expression tells us we need four consecutive integers that multiply together to equal 24.
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And the π-value will be the starting point, the largest of those four values.
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For smaller numbers like 24, we can try to solve this with a factor tree.
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24 equals two times 12, 12 equals two times six, and six is two times three.
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So we can say 24 equals two times three times four.
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But remember, for this problem, we need four consecutive integers that multiply together to equal 24 and not three.
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One, two, three, and four are four consecutive integers, and when multiplied together, they equal 24.
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Remember that our π-value is the largest value.
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If we let π equal four, 24 does equal four times three times two times one and confirms that π equals four.