A) \[{{14.2}^{14}}\]
B) \[{{13.2}^{14}}+1\]
C) \[{{13.2}^{14}}-1\]
D) None of these
Correct Answer: B
Solution :
[b] The general term \[{{T}_{r}}=(r-1){{\,}^{15}}{{C}_{r}},\] \[r=2,3,4,.....,15\] \[\therefore {{T}_{r}}=r{{\,}^{15}}{{C}_{r}}-{{\,}^{15}}{{C}_{r}}\] \[=15.{{\,}^{14}}{{C}_{r-1}}-{{\,}^{15}}{{C}_{r}}\,[\because \,r.{{\,}^{n}}{{C}_{r}}=n{{.}^{n-1}}{{C}_{r-1}}]\] \[\therefore \sum\limits_{r=2}^{15}{{{T}_{r}}=15\left[ ^{14}{{C}_{1}}{{+}^{14}}{{C}_{2}}+....+{{\,}^{14}}{{C}_{14}} \right]}\] \[-\left[ ^{15}{{C}_{2}}+{{\,}^{15}}{{C}_{3}}+...+{{\,}^{15}}{{C}_{15}} \right]\] \[=15[{{2}^{14}}-1]-[{{2}^{15}}-1-15]\] \[=(15-2){{2}^{14}}+1={{13.2}^{14}}+1\]
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