Explanation

(b)
Let the total number of persons in the room $= n$
$\therefore$ Total number of handshakes$= {\ ^{n}C}_{2} = 66$ (given)
$\Rightarrow \ \frac{n!}{2!(n - 2)!} = 66\ \ \Rightarrow \frac{n(n - 1)}{2} = 66$
$\Rightarrow \ \ n^{2} - n - 132 = 0$
$\ \Rightarrow \ (n - 12)(n + 11) = 0$
$\Rightarrow \ \ \ n = 12\ \ \ \ \ \ \ \ \ \lbrack\because n \neq - 11\rbrack$