A) \[{{2}^{20}}-{{2}^{5}}\]
B) \[\frac{20!}{5!15!}\]
C) \[\frac{20!}{5!15!}-1\]
D) \[\frac{20!}{5!15!}-\frac{15!}{5!10!}\]
E) \[\frac{15!}{5!10!}\]
Correct Answer: D
Solution :
Now, coefficient of \[{{x}^{15}}\]in\[{{(1+x)}^{20}}\] = coefficient of\[{{x}^{15}}\]in\[{{(1+x)}^{15}}{{(1+x)}^{5}}\] \[\Rightarrow \]\[^{20}{{C}_{15}}=\]coefficient of\[{{x}^{15}}\]in\[{{(}^{15}}{{C}_{0}}{{x}^{15}}{{+}^{15}}{{C}_{1}}{{x}^{14}}\] \[{{+}^{15}}{{C}_{2}}{{x}^{13}}{{+}^{15}}{{C}_{3}}{{x}^{12}}{{+}^{15}}{{C}_{4}}{{x}^{11}}{{+}^{15}}{{C}_{5}}{{x}^{10}})\] \[{{(}^{5}}{{C}_{0}}{{x}^{5}}{{+}^{5}}{{C}_{1}}{{x}^{4}}{{+}^{5}}{{C}_{2}}{{x}^{3}}{{+}^{5}}{{C}_{3}}{{x}^{2}}\] \[{{+}^{5}}{{C}_{4}}x{{+}^{5}}{{C}_{5}})\] \[\Rightarrow \]\[^{20}{{C}_{15}}{{=}^{15}}{{C}_{0}}{{.}^{5}}{{C}_{5}}{{+}^{15}}{{C}_{1}}{{.}^{5}}{{C}_{4}}{{+}^{15}}{{C}_{2}}{{.}^{5}}{{C}_{3}}\] \[{{+}^{15}}{{C}_{3}}{{.}^{5}}{{C}_{2}}{{+}^{15}}{{C}_{4}}{{.}^{5}}{{C}_{1}}{{+}^{15}}{{C}_{5}}^{5}{{C}_{0}}\] \[\Rightarrow \]\[^{15}{{C}_{0}}{{.}^{5}}{{C}_{5}}{{+}^{15}}{{C}_{1}}{{.}^{5}}{{C}_{4}}{{+}^{15}}{{C}_{2}}{{.}^{5}}{{C}_{3}}{{+}^{15}}{{C}_{3}}{{.}^{5}}{{C}_{2}}\] \[{{+}^{15}}{{C}_{4}}{{.}^{5}}{{C}_{1}}{{=}^{20}}{{C}_{15}}{{-}^{15}}{{C}_{5}}^{5}{{C}_{0}}\] \[=\frac{20!}{5!5!}-\frac{15!}{5!10!}\]
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