If the coefficients of $(2 r + 4)^{t h}, (r - 2)^{t h} t e r m s$ in the expansion of $(1 + x)^{18}$ are equal, find $r$ .

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Solution

A] Given $(1 + x)^{18}$
$(2 r + 3) + 1 =^{18} C_{2 r + 3} (x)^{2 r + 3}$
$^{T} (r - 3) + 1 =^{18} C_{2 r - 3} (x)^{r - 3}$
So,
$^{18} C_{2 r + 3} (x)^{2 r + 3} =^{18} C_{r - 3} (x)^{r - 3}$ $[^{n} C_{x} =^{n} c_{y} \Rightarrow x + y = n]$
= 2r + 3 + r - 3 = 18
= 3r = 18
$∴ r = 6$

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