Permutations and Combinations

If Find value of n. - Wired Faculty

Question

If $fraction numerator straight n factorial over denominator 2 factorial space left parenthesis straight n minus 2 right parenthesis factorial end fraction colon space fraction numerator straight n factorial over denominator 4 factorial space left parenthesis straight n minus 4 right parenthesis factorial end fraction equals space 2 space colon space 1$

Find value of n.

Solution

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

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

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fraction numerator 4 cross times 3 cross times 2 factorial cross times left parenthesis straight n minus 4 right parenthesis factorial over denominator 2 factorial left parenthesis straight n minus 2 right parenthesis left parenthesis straight n minus 3 right parenthesis cross times left parenthesis straight n minus 4 right parenthesis factorial end fraction equals 2 over 1

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

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(n - 2) (n - 3) = 6

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space space straight n squared minus 5 straight n plus 6 equals 6

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space space space straight n squared minus 5 straight n equals 0

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n (n - 5) = 0

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straight n equals 0 space not an element of space straight N comma space straight n space equals space 5 space element of space straight N

Hence, n = 5