Question

Find the value of $r$, if coefficient of $(2r+4{)}^{th}$ and $(r-2{)}^{th}$ terms in expansion of $(1+x{)}^{18}$ are equal:

• A. $5$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is C $6$

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,

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

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

On solving,

$2r+3=r-3$

$r=-6$ which is not possible.

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

$3r=18$

$r=6$

Hence, C is the correct option.