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Sum of Binomial Coefficients of Odd Numbered Terms

Determine n i...

Question

Determine n if
(i) 2nC3:nC3 =12:1

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Solution

$C presuperscript 2 n end presuperscript subscript 3 colon C presuperscript n subscript 3 equals 12 colon 1 c a n space b e space w r i t t e n space a s comma fraction numerator C presuperscript 2 n end presuperscript subscript 3 over denominator C presuperscript n subscript 3 end fraction equals 12 over 1 H e r e space w e space k n o w space C presuperscript 2 n end presuperscript subscript 3 equals fraction numerator left parenthesis 2 n right parenthesis factorial over denominator left parenthesis 2 n minus 3 right parenthesis factorial 3 factorial end fraction space a n d space C presuperscript n subscript 3 equals fraction numerator n factorial over denominator 2 factorial left parenthesis n minus 2 right parenthesis factorial end fraction equals greater than fraction numerator fraction numerator left parenthesis 2 n right parenthesis factorial over denominator left parenthesis 2 n minus 3 right parenthesis factorial 3 factorial end fraction over denominator fraction numerator n factorial over denominator 2 factorial left parenthesis n minus 2 right parenthesis factorial end fraction end fraction equals 12 equals greater than fraction numerator left parenthesis 2 left parenthesis 2 n minus 1 right parenthesis left parenthesis 2 n minus 2 right parenthesis right parenthesis over denominator 3 left parenthesis n minus 1 right parenthesis end fraction equals 12 space space left square bracket space S i n c e space fraction numerator left parenthesis 2 n right parenthesis factorial over denominator left parenthesis 2 n minus 3 right parenthesis factorial end fraction equals left parenthesis left parenthesis 2 n right parenthesis left parenthesis 2 n minus 1 right parenthesis left parenthesis 2 n minus 2 right parenthesis space a n d space fraction numerator n factorial over denominator left parenthesis n minus 2 right parenthesis factorial end fraction equals n left parenthesis n minus 1 right parenthesis space a n d space a l s o space fraction numerator 3 factorial over denominator 2 factorial end fraction equals 3 right square bracket equals greater than fraction numerator 2 left parenthesis 2 n minus 1 right parenthesis 2 left parenthesis n minus 1 right parenthesis over denominator 3 left parenthesis n minus 1 right parenthesis end fraction equals 12 equals greater than space 4 left parenthesis 2 n minus 1 right parenthesis equals 36 equals greater than space 2 n equals 10 equals greater than space n equals 5 H o p e space t h i s space h e l p s space colon right parenthesis$

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Determine n if

(i) (ii)

Determine n if

(i) (ii)

^{2 n} C_{3} :^{n} C_{3} = 12 : 1

^{2 n} C_{3} :^{n} C_{3} = 11 : 1

n

^{2 n} C_{3} :^{n} C_{3} = 12 : 1

n

^{2 n} C_{3} :^{n} C_{3} = 12 : 1

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