JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Self Evaluation Test - Binomial Theorem

If 'n' is positive integer and three consecutive coefficient in the expansion of \[{{(1+x)}^{n}}\] are in the ratio 6 : 33 : 110, then n is equal to:

A) 9

B) 6

C) 12

D) 16

Correct Answer: C

Solution :

[c] Let the consecutive coefficient of

\[{{(1+x)}^{n}}\] are \[^{n}{{C}_{r-1}},{{\,}^{n}}{{C}_{r}},{{\,}^{n}}{{C}_{r+1}}\]

From the given condition, \[^{n}{{C}_{r-1}}:{{\,}^{n}}{{C}_{r}}:{{\,}^{n}}{{C}_{r+1}}=6\]

\[:33:110\]

Now \[^{n}{{C}_{r-1}}:{{\,}^{n}}{{C}_{r}}=6:33\]

\[\Rightarrow \frac{n!}{(r-1)!(n-r+1)!}\times \frac{r!(n-r)!}{n!}=\frac{6}{33}\]

\[\Rightarrow \frac{r}{n-r+1}=\frac{2}{11}\Rightarrow 11r=2n-2r+2\]

\[\Rightarrow 2n-13r+2=0...(i)\]

and \[^{n}{{C}_{r}}:{{\,}^{n}}{{C}_{r+1}}=33:110\]

\[\Rightarrow \frac{n!}{r!(n-r)!}\times \frac{(r+1)!(n-r-1)}{n!}=\frac{33}{110}=\frac{3}{10}\]

\[\Rightarrow \frac{(r+1)}{n-r}=\frac{3}{10}\Rightarrow 3n-13r-10=0..(ii)\]

Solving (i) & (ii), we get \[n=12\]