If the coefficient of $(2 r + 3)^{t h}$ and $(r - 1)^{t h}$ terms in the expansion of $(1 + x)^{18}$ are equal then find the value of $r$ .

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Solution

The correct option is C $6$
$T_{2 r + 2 + 1} =^{18} C_{2 r + 2} (1)^{18 - 2 r - 2} x^{2 r + 2}$
$T_{r - 2 + 1} =^{18} C_{r - 2} (1)^{18 - r + 2} x^{r - 2}$
$⟹^{18} C_{2 r + 2} =^{18} C_{r - 2}$
$⟹ 2 r + 2 = r - 2$
$r = - 4$ which is not possible
Therefore , $2 r + 2 = 18 - r + 2$
$⟹ 3 r = 18$
$⟹ r = 6$

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