Explanation

(c)
Given, $3\sin^{2}{x - 7\sin{x + 2 = 0}}$
$\left( 3\sin{x - 1} \right)\left( \sin{x - 2} \right) = 0$
$\Rightarrow \ \sin{x = \frac{1}{3}}\ or\ 2$
$\Rightarrow \ \sin{x = \frac{1}{3}\ \ \ \ \lbrack\because\sin{x \neq 2}\rbrack}$
Let $\sin^{- 1}{\frac{1}{3} = \alpha,\ 0 < \alpha < \frac{\pi}{2}}$, then $\alpha,\ \pi - \alpha,\ 2\pi + \alpha,\ 3\pi - \alpha,\ 4\pi + \alpha,\ 5\pi - \alpha$ are the solution in $\lbrack 0,\ 5\pi\rbrack$
Hence, required number of solutions are 6