Question:
If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r is equal to
(a) 7
(b) 8
(c) 9
(d) 10
Solution:
(c) 9
Coefficients of the $r$ th and $(r+4)$ th terms in the given expansion are ${ }^{20} C_{r-1}$ and ${ }^{20} C_{r+3}$.
Here,
${ }^{20} C_{r-1}={ }^{20} C_{r+3}$
$\Rightarrow r-1+r+3=20 \quad\left[\because\right.$ if ${ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y$ or $\left.x+y=n\right]$
$\Rightarrow r=2$ or $2 r=18$
$\Rightarrow r=9$