find the number of ways 1 can put 3 balls numbered 1,2,3 in three boxes labelled a,b,c such that at the most one box is empty

We have three diff balls and three diff box .

Now at most one box is empty - so there are two cases

1. None box is empty

2. One box is empty

therefore , if none box is empty number of ways =3x2x1 =6

if one box is empty , we can choose that box in 3 ways and the 3 balls can be put in two boxes in 3P2 ways .

so number of ways in case 2= 3x3p2= 3x6=18

therefore total number of ways = 18+6=24