TITLE: Placing 3 identical bananas into 10 numbered distinct boxes
QUESTION [0 upvotes]: I am a little bit confused on how to count the number of ways to distribute 3 identical bananas into 10 numbered boxes. Where we can have more than one banana in each box.
My teacher divided the problem into 4 cases.
case 1: 1 banana in each box 
case 2: 2 bananas then 1 banana 
case 3: 1 banana then 2 bananas 
case 4: 3 bananas in each box.
Then each case could be solved by combination.
My first question is why are case 2 and case 3 different aren't we double counting ? isn't one of them already takes into consideration all possible ways to distribute 2 bananas and 1 banana.
I tried my own way to solve the problem, however I know that it is false since I am not getting an integer number. The way a proceeded is that we take the first banana, there are 10 different boxes to put them in so that's 10, then we take the second banana and also 10 possible boxes to place the banana then finally we have the third bananas which also have 10 possible boxes. 
So by the rule of product we have 10*10*10. we have 3 identical bananas so we must divide by 3!
I know that my answer is wrong (since it is not an integer) but I don't see why it is wrong, for me the logic makes sense but I could be missing something.
I am struggling to understand the logic behind this problem.

REPLY [1 votes]: Your answer is wrong because when two bananas go into the same box, it's not correct to divide those combinations by $6$.  If that's the approach you use, then it's essential to divide into cases along the lines of the solution presented by your teacher.
But for this sort of problem I prefer the approach often known as "stars and bars."  You have $9$ bars representing the dividers between the $10$ boxes.  You have $3$ stars representing the $3$ bananas.  There are $\binom{12}{3}=220$ ways to arrange the stars and bars in a line, and each different arrangement yields a unique division of the bananas into boxes.
{"set_name": "stack_exchange", "score": 0, "question_id": 3395389}