Find the value of r,


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.


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,

${ }^{18} C_{2 r+3}={ }^{18} C_{r-3}$

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

$3 r=18 \quad \therefore \quad r=6$

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