A) 7, 5
B) 5, 4
C) 4, 2
D) 3, 6
Correct Answer: C
Solution :
\[{{(1+x)}^{n}}=1{{+}^{n}}{{C}_{1}}x{{+}^{n}}{{C}_{2}}{{x}^{2}}{{+}^{n}}{{C}_{3}}{{x}^{3}}+....\] coefficient of second term\[{{=}^{n}}{{C}_{1}}\] coefficient of third term\[{{=}^{n}}{{C}_{2}}\] coefficient of fourth term\[{{=}^{n}}{{C}_{3}}\] \[\because \]These are in AP. \[\Rightarrow \]\[2\frac{n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{3}\] \[\Rightarrow \] \[{{n}^{2}}-n=\frac{3n+({{n}^{2}}-n)(n-2)}{3}\] \[\Rightarrow \] \[3{{n}^{2}}-3n=3n+{{n}^{3}}-3{{n}^{2}}+2n\] \[\Rightarrow \] \[{{n}^{3}}-6{{n}^{2}}+8n=0\] \[\Rightarrow \] \[{{n}^{2}}-6n+8=0\] \[\Rightarrow \] \[(n-4)(n-2)=0\] \[\Rightarrow \] \[n=4,2\]
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