Question

If the binomial coeff of $(2r+4{)}^{th}$ term and the $(r-2{)}^{th}$ term in the expansion of $(1+x{)}^{18}$ are equal, find the value of $r$ :

• A. $14$
• B. $6$
• C. $12$
• D. $9$

Answer 1

Answer: (B)

$6$

Solution:

We know that in the coefficient of $r^{th}$ term in expansion of $(1+x{)}^n={}^n{C}_{r-1}$

$\therefore$ coefficients of $(2r+4{)}^{th}$ and $(r-2{)}^{th}$ in the expansion $(1+x{)}^{18}$ are ${}^{18}{C}_{2r+3}$ and ${}^{18}{C}_{r-3}.$

Now,

${}^{18}{C}_{2r+3}={}^{18}{C}_{r-3}$

or, $2r+3=r-3$ or, $2r+3+r-3=18$ $(\because {}^n{C}_r={}^{n}Cs$ then $r=s$ or $r+s=n)$

On solving,

or $2r+3=r-3$

or, $r=-6$ which is not possible.

$2r+3+r-3=18$

or, $3r=18$

or, $r=6$

Hence, B is the correct option.