Question

If the coefficient of $(2r+4)$th term and $(r-2)$th terms in the expansion of ${(1+x)}^{18}$ are equal then $r=$

• A. $9$
• B. $4$
• C. $6$
• D. $3$

Answer 1

The correct option is C $6$

Given expression is $(1+x{)}^{18}$

Now, $(2r+4{)}^{th}$ term i.e $T_{(2r+3)+1}$

$\therefore T_{(2r+3)+1}{=}^{18}{C}_{2r+3}(x{)}^{2r+3}$

And, $(r-2{)}^{th}$ term i.e $T_{(r-3)+1}$

$\therefore T_{(r-3)+1}{=}^{18}{C}_{r-3}(x{)}^{r-3}$

According to the question

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

As we know

${}^n{C}_x{=}^n{C}_y\Rightarrow x+y=n$

So,

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

$3r=18$

$\therefore r=6$