A) 66
B) 36
C) 60
D) None of these
Correct Answer: A
Solution :
Number of ways = coefficient of \[{{x}^{15}}\] in the expansion \[(1+x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+{{x}^{5}})\] \[(1+x+{{x}^{2}}+.......+{{x}^{10}})\] \[(1+x+{{x}^{2}}+......+{{x}^{15}})\] \[(1+x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+{{x}^{5}})(1+x+{{x}^{2}}+.....+{{x}^{10}})\] \[(1+x+{{x}^{2}}+...+{{x}^{15}})=(1-{{x}^{6}}-{{x}^{11}})(1+{{\,}^{3}}{{C}_{1}}x+{{\,}^{4}}{{C}_{2}}{{x}^{2}}\] \[+......+{{\,}^{6}}{{C}_{4}}{{x}^{4}}+{{\,}^{11}}{{C}_{9}}{{x}^{9}}+{{\,}^{17}}{{C}_{15}}{{x}^{15}}+............)\] \[=.......+.......+{{x}^{15}}({{-}^{11}}{{C}_{9}}-{{\,}^{6}}{{C}_{4}}+{{\,}^{17}}{{C}_{15}})\] \[=.......+......+{{x}^{15}}(-55-15+136)\]\[={{x}^{15}}\times 66\] \ Coefficient of\[{{x}^{15}}=66\].
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