Question
Tue March 03, 2015 By: Surajsubedi14

, Question no 88945, Mathematics

Expert Reply
Vijaykumar Wani
Tue March 03, 2015
begin mathsize 18px style open parentheses x plus y close parentheses to the power of n equals x to the power of n plus C presuperscript n subscript 1 x to the power of n minus 1 end exponent y plus C presuperscript n subscript 2 x to the power of n minus 2 end exponent y squared plus horizontal ellipsis horizontal ellipsis plus C presuperscript n subscript r x to the power of n minus r end exponent y to the power of r plus horizontal ellipsis horizontal ellipsis plus C presuperscript n subscript n minus 1 end subscript x y to the power of n minus 1 end exponent plus y to the power of n open parentheses x to the power of 4 minus 1 over x cubed close parentheses to the power of 15 equals open square brackets x to the power of 4 plus open parentheses negative x to the power of negative 3 end exponent close parentheses close square brackets to the power of 15 equals open parentheses x to the power of 4 close parentheses to the power of 15 plus C presuperscript 15 subscript 1 open parentheses x to the power of 4 close parentheses to the power of 14 open parentheses negative x to the power of negative 3 end exponent close parentheses plus horizontal ellipsis horizontal ellipsis plus C presuperscript 15 subscript r open parentheses x to the power of 4 close parentheses to the power of 15 minus r end exponent open parentheses negative x to the power of negative 3 end exponent close parentheses to the power of r plus horizontal ellipsis horizontal ellipsis plus C presuperscript 15 subscript 14 open parentheses x to the power of 4 close parentheses open parentheses negative x to the power of negative 3 end exponent close parentheses to the power of 14 plus open parentheses negative x to the power of negative 3 end exponent close parentheses to the power of 15  T h e space g e n e r a l space t e r m space o f space t h e space e x p a n s i o n space i s space g i v e n space b y C presuperscript 15 subscript r open parentheses x to the power of 4 close parentheses to the power of 15 minus r end exponent open parentheses negative x to the power of negative 3 end exponent close parentheses to the power of r equals C presuperscript 15 subscript r x to the power of 60 minus 4 r end exponent open parentheses negative 1 close parentheses to the power of r x to the power of negative 3 r end exponent space equals C presuperscript 15 subscript r open parentheses negative 1 close parentheses to the power of r x to the power of 60 minus 7 r end exponent....... left parenthesis i right parenthesis  E q u a t i n g space t h e space p o w e r s space o f space x space t o space 32 space i n space e q u a t i o n space left parenthesis i right parenthesis space w e space g e t comma 60 minus 7 r equals 32 rightwards double arrow r equals 4  T a k e space r equals 4 space i n space t h e space e q u a t i o n space left parenthesis i right parenthesis space w e space g e t C presuperscript 15 subscript 4 open parentheses negative 1 close parentheses to the power of 4 x to the power of 60 minus 28 end exponent equals fraction numerator 15 factorial over denominator 11 factorial 4 factorial end fraction x to the power of 32 equals 1365 x to the power of 32  S o space c o e f f i c i e n t space o f x to the power of 32 equals 1365. space  E q u a t i n g space t h e space p o w e r s space o f space x space t o space minus 17 space i n space e q u a t i o n space left parenthesis i right parenthesis space w e space g e t comma 60 minus 7 r equals negative 17 rightwards double arrow r equals 11  T a k e space r equals 11 space i n space t h e space e q u a t i o n space left parenthesis i right parenthesis space w e space g e t C presuperscript 15 subscript 11 open parentheses negative 1 close parentheses to the power of 11 x to the power of 60 minus 77 end exponent equals negative fraction numerator 15 factorial over denominator 11 factorial 4 factorial end fraction x to the power of negative 17 end exponent equals negative 1365 x to the power of negative 17 end exponent  S o space c o e f f i c i e n t space o f x to the power of negative 17 end exponent equals negative 1365. space end style
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