Total number of ways in which 10 balls can be placed in 4 boxes =$4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4$

$\Rightarrow 4^{10}$

$\Rightarrow 2^{20}$

 Exactly two box contain $2,3$ balls 

$\Rightarrow {}^{4}C_2\times 2\times {}^{10}C_2\times {}^{8}C_3\times 2^5$

Probability $=\frac{{}^{4}C_2\times 2\times {}^{10}C_2\times {}^{8}C_2\times 2^5}{2^{20}}$

$=\frac{945}{2^{10}}$