The Binomial theorem related question as well as part of the solution in attached above. Please arrange to send the remaining solution step by step. Unless you provide the complete end solution the very purpose of getting answer or solution asked from uou is defeated. So please take care this point in sending the solution of each and every problem in future..

Asked by sudhanshubhushanroy | 14th Nov, 2017, 09:40: AM

Expert Answer:

begin mathsize 16px style sum from straight gamma equals straight m to straight capital gamma of open parentheses straight n minus straight r plus 1 close parentheses space to the power of straight r straight C subscript straight m space
equals open parentheses straight n minus straight m plus 1 close parentheses space to the power of straight m straight C subscript straight m space plus open parentheses straight n minus straight m close parentheses space to the power of straight m plus 1 end exponent straight C subscript straight m space plus open parentheses straight n plus straight m close parentheses space to the power of straight m plus 2 end exponent straight C subscript straight m plus... plus open parentheses 2 close parentheses space to the power of straight n minus 1 end exponent straight C subscript straight m space plus open parentheses 1 close parentheses space to the power of straight n straight C subscript straight m
write space in space reverse space
equals to the power of straight n straight C subscript straight m space plus open parentheses 2 close parentheses space to the power of straight n minus 1 end exponent straight C subscript straight m plus open parentheses 3 close parentheses space to the power of straight n minus 2 end exponent straight C subscript straight m space plus..... plus open parentheses straight n minus straight m close parentheses space to the power of straight m plus 1 end exponent straight C subscript straight m space plus open parentheses straight n minus straight m plus 1 close parentheses space to the power of straight m straight C subscript straight m
seperate space out space terms
equals to the power of straight n straight C subscript straight m space plus space to the power of straight n minus 1 end exponent straight C subscript straight m plus space to the power of straight n minus 2 end exponent straight C subscript straight m space plus....... space plus to the power of straight m straight C subscript straight m space space space space space space left parenthesis equals space to the power of straight n plus 1 end exponent straight C subscript straight m plus 1 end subscript space right parenthesis
plus space to the power of straight n minus 1 end exponent straight C subscript straight m plus to the power of straight n minus 2 end exponent straight C subscript straight m space plus........... space plus space to the power of straight m straight C subscript straight m space space space space end subscript space left parenthesis equals space to the power of straight n straight C subscript straight m plus 1 end subscript space right parenthesis
plus to the power of straight n minus 2 end exponent straight C subscript straight m space plus................... space plus to the power of straight m straight C subscript straight m space end subscript space space space left parenthesis equals space to the power of straight n minus 1 end exponent straight C subscript straight m plus 1 end subscript space right parenthesis
plus to the power of straight n minus 3 end exponent straight C subscript straight m space plus................... space plus to the power of straight m straight C subscript straight m space space space left parenthesis equals space to the power of straight n minus 2 end exponent straight C subscript straight m plus 1 end subscript space right parenthesis
plus.................. so space on space till space to the power of straight m straight C subscript straight m space space space left parenthesis space to the power of straight m plus 1 end exponent straight C subscript straight m plus 1 end subscript space right parenthesis

so space to the power of straight n plus 1 end exponent straight C subscript straight m plus 1 end subscript plus to the power of straight n straight C subscript straight m plus 1 end subscript plus to the power of straight n minus 1 end exponent straight C subscript straight m plus 1 end subscript plus.............. plus to the power of straight m plus 1 end exponent straight C subscript straight m plus 1 end subscript equals to the power of straight n plus 2 end exponent straight C subscript straight m plus 2 end subscript
end style
PS: Our purpose is that a student should follow the lead and give things a try so that in future they can solve such examples. 

Answered by Arun | 7th Dec, 2017, 10:11: AM