Question

Given positive integers $r >1, n > 2$ and the coefficient of $(3r)th$ and $ (r + 2)th $ terms in the binomial expansion of $(1 + x)^2n$ are equal. Then,

• A. n=2r
• B. n=2r+1
• C. n=3r
• D. None of these

Answer 1

The Correct Option is A

In the expansion $(1+x)^{2n},t_{3r} = \, ^{2n}C_{3r-1} (x)^{3r-1}$
and $\hspace20mm \, t_{r+2}= \, ^{2n}C_{r+1} (x) ^{r+1}$
Since, binomial coefficients of t$_3r$ and t$_{r+ 2}$ are equal
$\Rightarrow \, \, \, \, \, \, \, \, 3r-1=r+1 \, or \, 2n=(3r-1)+(r+1)$
$\Rightarrow \, \, \, \, \, \, \, \, $ 2r=2 $ \, or \, \, \, \, \, \, \, \, $ 2n = 4r
$\Rightarrow \, \, \, \, \, \, \, \, $ r = 1 $ \, or \, \, \, \, \, \, \, \, $ n=2r
But $ \, \, \, \, \, \, \, \, $ r >1
$\therefore \, \, $We take, n=2r