Question:
Find the value of $r$, if the coefficients of $(2 r+4)^{\text {th }}$ and $(r-2)^{\text {th }}$ terms in the expansion of $(1+x)^{18}$ are equal.
Solution:
Given $(1+x)^{18}$
Now, $(2 r+4)^{\text {th }}$ term,
That is $T_{(2 r+3)+1}$
$T_{(2 r+3)+1}={ }^{18} C_{2 r+3}(x)^{2 r+3}$
And $(r-2)^{\text {th }}$ term, that is $T_{(r-3) \mid+1}$
$T_{(r-3)+1}={ }^{18} C_{r-3} x^{-3}$
Now according to the question,
${ }^{18} C_{2 r+3}={ }^{18} C_{r-3}$
$2 r+3+r-3=18$
$3 r=18 \quad \therefore \quad r=6$