(c) Let the boxes be marked as $A,\;B,\;C$. We have to ensure that no box remains empty and in all five balls have to put in. There will be two possibilities.

$(i)$ Any two containing one and ${3^{rd}}$ containing $3.$

$A$ $(1)$ $B$ $(1)$ $C$ $(3)$

$^5{C_1}{.^4}{C_1}{.^3}{C_3} = 5\;.\;4\;.\;1 = 20$.

Since the box containing $3$ balls could be any of the three boxes $A,\;B,\;C$.

Hence the required number is = $20 \times 3 = 60$.

$(ii)$ Any two containing $2$ each and ${3^{rd}}$ containing $1$.

$A$ $(2)$ $B$ $(2)$ $C$ $(1)$

$^5{C_2}{.^3}{C_2}{.^1}{C_1} = 10 \times 3 \times 1 = 30$

Since the box containing $1$ ball could be any of the three boxes $A,\;B,\;C$.

Hence the required number is = $30 \times 3 = 90$.

Hence total number of ways are = $60 + 90 = 150$.