Say for instance a food stall keeps a record of how many complaints it receives during its 4 day work week. Complaints are classified as Mon, Tues, Wed or Thurs, depending on the day during the week which the complaint was received. How many different classifications of 10 complaints can there be in one work week?
My working:
By adding the total partitions of each possible parts:
Partitions of 10 into 4 parts: $$= 9$$
Partitions of 10 into 3 parts x 2 (as there are 2 possible locations where this could start if considering consecutive day order):
$$ = 8 \times 2 $$
Partitions of 10 into 2 parts x 3:
$$ = 5 \times 3$$
Partition of 10 into 1 part x 4 (as 4 days of the week where all the complaints were reported):
$$ = 1 \times 4$$
Totalling: $$9 + 16 + 15 + 4$$ $$=44$$
Is this correct? Logically it sounds correct but I'm not sure whether I am over/undercounting/even on the right track!
A really good way to do partitions is to use "stars and bars"
In this case you would use 10 stars representing complaints and 3 bars representing partition boundaries
So that
**||*****|***represents (Mon, Tues, Wed, Thur) = $(2,0,5,3)$Number of partitions = number of 13-character strings containing 10 stars and 3 bars.
To count these you just need to select the positions that you place the 3 bars and then you just fill the rest with stars.
so the answer is $$\binom {13}{3}=\frac{13 \times 12 \times 11}{3 \times 2 \times 1} =286 $$