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Permutations And Combinations

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$WiredFaculty$ $space space space space space space space fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times........... space left parenthesis 2 straight n minus 1 right parenthesis right square bracket 2 to the power of straight n$

Solution

L.H.S. = $fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction equals fraction numerator 1 times 2 times 3 times 4 times 5 times..... left parenthesis 2 straight n minus 1 right parenthesis space left parenthesis 2 straight n right parenthesis over denominator straight n factorial end fraction$
= $fraction numerator open square brackets 1 times 3 times 5 times....... left parenthesis 2 straight n minus 1 right parenthesis close square brackets space left square bracket 2 times 4 times space.......... left parenthesis 2 straight n right parenthesis over denominator straight n factorial end fraction$
= $fraction numerator left square bracket 1 times 3 times 5 times space..... left parenthesis 2 straight n minus 1 right parenthesis right square bracket space cross times space 2 to the power of straight n cross times left square bracket 1 times 2 times......... straight n right square bracket over denominator straight n factorial end fraction$
= $space space space fraction numerator left square bracket 1 times 3 times 5 times........ left parenthesis 2 straight n minus 1 right parenthesis space cross times space 2 to the power of straight n space cross times space straight n factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times....... space left parenthesis 2 straight n minus 1 right parenthesis right square bracket space 2 to the power of straight n$
= R.H.S.
∴ L.H.S. = R.H.S.
Hence, $space space fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times....... left parenthesis 2 straight n minus 1 right parenthesis right square bracket 2 to the power of straight n$

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