$6$ letters are posted in three letter boxes. The number of ways posting the letters when no letter box remains empty.

$6$ letters have to be put is $3$ post boxes.

and $(A,B,C)\ge 1$

Case (i): $3+2+1$

We select one post box and put $3$ of the $6$ letters in that post box, then select another post box and put $2$ letters(of the $3$ remaining)

no. of ways$={}^3{C}_1\times {}^6{C}_3\times {}^2{C}_1\times {}^3{C}_2=360$

Case (ii): $2+2+2$

We select each box and put $2$ letters each in each box.

No. of ways$\Rightarrow {}^3{C}_1\times {}^6{C}_2\times {}^2{C}_1\times {}^4{C}_2\times \frac{1}{3}$

Total no. of ways$=360+180=540$.