A) \[^{17}{{C}_{2}}\]
B) \[^{17}{{C}_{3}}\]
C) \[^{18}{{C}_{2}}\]
D) \[^{18}{{C}_{3}}\]
Correct Answer: D
Solution :
Given function is \[f(y)=1-(y-1)+{{(y-1)}^{2}}-{{(y-1)}^{3}}\] \[+......-{{(y-1)}^{17}}\] In the expansion of \[{{(y-1)}^{n}}\] \[{{T}_{r+1}}{{=}^{n}}{{C}_{r}}{{y}^{n-r}}{{(-1)}^{r}}\] coeff of \[{{y}^{2}}\]in\[{{(y-1)}^{2}}{{=}^{2}}{{C}_{0}}\] coeff of\[{{y}^{2}}\]in\[{{(y-1)}^{3}}={{-}^{3}}{{C}_{1}}\] coeff of\[{{y}^{2}}\]in\[{{(y-1)}^{4}}{{=}^{4}}{{C}_{2}}\] so, coeff of term wise is \[2{{C}_{0}}{{+}^{3}}{{C}_{1}}+4{{C}_{2}}+5{{C}_{3}}+.........{{+}^{17}}{{C}_{15}}\] \[=1{{+}^{3}}{{C}_{1}}{{+}^{4}}{{C}_{2}}{{+}^{5}}{{C}_{3}}+.........{{+}^{17}}{{C}_{15}}\] \[={{(}^{3}}{{C}_{0}}{{+}^{3}}{{C}_{1}}){{+}^{4}}{{C}_{2}}{{+}^{5}}{{C}_{3}}+..........{{+}^{17}}{{C}_{15}}\] \[{{=}^{4}}{{C}_{1}}{{+}^{4}}{{C}_{2}}{{+}^{5}}{{C}_{3}}+..........{{+}^{17}}{{C}_{15}}\] \[{{=}^{5}}{{C}_{2}}{{+}^{5}}{{C}_{3}}+..........{{+}^{17}}{{C}_{15}}\] \[{{=}^{18}}{{C}_{15}}{{=}^{18}}{{C}_{3}}\]
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