Averages part V

[BMAD]

I walked by a holiday display, and noticed there were nine straw baskets, each one having a whole number of apples, at most 9, and possibly none. Then I also noticed the mean number of apples was 4, the median was 4, and the mode was 2. Is this possible, and if so, how many solutions are there?

(not counting the positions of the baskets)

[Rainman]

There are 34 possibilities. We can think of the solutions as all 9-digit numbers satisfying the following criteria:

• The digits are in ascending order (since we don't count the positions of the basket we might as well order them)
• The number can start with 0
• The single most common digit is 2 (mode is 2)
• The middle digit is 4 (median is 4)
• The digit sum is 36 (average is 4)

These are all the solutions: 012245679, 022244589, 022244679, 022244688, 022244778, 022245579, 022245588, 022245669, 022245678, 022246677, 122244579, 122244588, 122244669, 122244678, 122245569, 122245578, 122245668, 122245677, 222244479, 222244488, 222244569, 222244578, 222244668, 222244677, 222245559, 222245568, 222245577, 222245667, 222344559, 222344568, 222344577, 222344667, 222345567, 222445566