so there are 3 types of boxes, 1kg, 3kg and 5kg. two conditions are there: 1) not all types boxes are used 2) atleast one of each type box is used.
considering condition 1
observation: salesman cant use six boxes of 5kg (total exceeds 25kg). so total no of 5kg box is less than or equal to five.
case 1: five boxes of 5kg total sums upto 25kg, so only possibility is zero 3kg box and zero 1 kg box (solution 1)
case 2: four boxes of 5kg total sums upto 20kg, so maximum number of 3kg boxes cant be more than one (20+(2)*3>25). possibilities are, one 3kg boxes and two 1kg boxes (solution 2) zero 3kg boxes and five 1kg boxes (solution 3)
case 3: three boxes of 5kg total sums upto 15kg, so maximum number of 3kg boxes cant be more than three (15+(4)*3>25). possibilities are, three 3kg boxes and one 1kg boxes (solution 4) two 3kg boxes and four 1kg boxes (solution 5) note: if number of 3kg box is less than two, then we need more than or equal to seven 1kg boxes, that exceeds total no of boxes above 10.
case 4: two boxes of 5kg total sums upto 10kg, so maximum number of 3kg boxes cant be more than five (10+(6)*3>25). possibilities are, five 3kg boxes and zero 1kg boxes (solution 6) four 3kg boxes and three 1kg boxes (solution 7) note: if number of 3kg box is less than four, then we need more than or equal to six 1kg boxes, that exceeds total no of boxes above 10.
case 5: one box of 5kg total sums upto 5kg, so maximum number of 3kg boxes cant be more than six (5+(7)*3>25). possibilities are, six 3kg boxes and two 1kg boxes (solution 8)
note: if number of 3kg box is less than six, then we need more than or equal to five 1kg boxes, that exceeds total no of boxes above 10.
case 6: zero box of 5kg total sums upto 0kg, so maximum number of 3kg boxes cant be more than eight (0+(9)*3>25). possibilities are, eight 3kg boxes and one 1kg boxes (solution 9)
note: if number of 3kg box is less than eight, then we need more than or equal to four 1kg boxes, that exceeds total no of boxes above 10.
solutions are:
- five 5kg boxes, zero 3kg boxes and zero 1kg boxes
- four 5kg boxes, zero 3kg boxes and five 1kg boxes
- four 5kg boxes, one 3kg boxes and two 1kg boxes
- three 5kg boxes, three 3kg boxes and one 1kg boxes
- three 5kg boxes, two 3kg boxes and four 1kg boxes
two 5kg boxes, five 3kg boxes and zero 1kg boxes
two 5kg boxes, four 3kg boxes and three 1kg boxes
- one 5kg boxes, six 3kg boxes and two 1kg boxes
- zero 5kg boxes, eight 3kg boxes and one 1kg boxes
if the scenario of at least one of each type box being used is considered then solutions 3,4,5,7 and 8 is taken.