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Permutations And Combinations

Question

Wired Faculty App

Determine n if:

$straight C presuperscript 2 straight n end presuperscript subscript 3 colon straight C presuperscript straight n subscript 3 space equals space 12 colon 1$

Solution

straight C presuperscript 2 straight n end presuperscript subscript 3 space colon space straight C presuperscript straight n subscript 3 space equals space 12 colon 1

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fraction numerator 2 straight n factorial over denominator left parenthesis 2 straight n minus 3 right parenthesis factorial space 3 factorial end fraction colon fraction numerator straight n factorial over denominator left parenthesis straight n minus 3 right parenthesis factorial space 3 factorial end fraction space equals space 12 colon 1

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fraction numerator 2 straight n left parenthesis 2 straight n minus 1 right parenthesis space left parenthesis 2 straight n minus 2 right parenthesis space left parenthesis 2 straight n minus 3 right parenthesis factorial over denominator left parenthesis 2 straight n minus 3 right parenthesis factorial space 3 factorial end fraction cross times fraction numerator left parenthesis straight n minus 3 right parenthesis factorial space cross times space 3 factorial over denominator straight n left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 2 right parenthesis space left parenthesis straight n minus 3 right parenthesis factorial end fraction space equals space 12 over 1

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fraction numerator 2 straight n left parenthesis 2 straight n minus 1 right parenthesis left parenthesis 2 straight n minus 2 right parenthesis over denominator straight n left parenthesis straight n minus 1 right parenthesis left parenthesis straight n minus 2 right parenthesis end fraction equals 12

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fraction numerator 4 left parenthesis 2 straight n minus 1 right parenthesis over denominator left parenthesis straight n minus 2 right parenthesis end fraction space equals space 12

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8n - 4 = 12n - 24

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4n = 20

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n = 5

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