Twelvefold Way/Combinatorics - Determining Balls vs Urns and Labelled vs Unlabelled I'm having a lot of trouble deciding when to call an object in a problem a Ball vs an Urn (a box). Once that's decided, I'm then unsure when to call each of them labelled vs unlabelled.
Current understanding and thought process
My current understanding is one ball can only go into one urn. An urn cannot go into a ball.
When deciding if an object is a ball vs an urn, I ask myself: Does it make literal sense to put X into Y. If it does, I call X the ball and Y the urn.
Then when I decide if a ball or urn is labelled vs unlabelled, I ask myself: Can we distinguish one object from another? If yes, it's labelled.
Some example problems


*

*Knowing that from 10 people we can choose a committee in 210 different ways, in how many ways can we chose the same committee if we have 12 people?



*Consider that you have a bowl of generic fruits (different amounts of the same fruit do not make a difference). How many different fruit salads can you make using a certain amount of ingredients per salad (a certain number of fruits that you are to put in each salad), if you know that excluding a fruit from the initial bowl will give you 1330 ways of making fruit salads each with as many ingredients, while excluding a fruit and using an ingredient less, will give you 5985 different fruit salads.

Questions vs thought process
The first question I first struggled with identifying whether a committee position was distinguishable from others? (are the roles different e.g. the president vs secretary)
The second question says that the fruit bowl could contain more than one of the same fruit so they're indistinguishable meaning something is unlabelled. I'm guessing the bowls here are the other object in the question. To find out which is the ball vs the urn, I ask 'Does it make literal sense to put fruit into fruit bowls'. It does, so I consider the fruit to be balls and the urns to be the fruit bowls. The balls are unlabelled, the urns I'm still not sure here but I guessed unlabelled because how can we tell one bowl from another? Also, to classify as a fruit salad each bowl must have at least one piece of fruit. Therefore I'd say the question is unlabelled balls, unlabelled urns and more than one ball per urn but apparently the question is unlabelled balls, labelled urns and more than one ball per urn.
Could someone help improve questions I can ask myself to understand the question in terms of balls vs urns and labelled vs unlabelled or help guide me?
 A: This answer is rather late, but I just saw it because I've also been trying to find a systematic way of putting combinatorial problems into definite categories.
See https://en.wikipedia.org/wiki/Twelvefold_way for one way of categorizing counting problems of placing a set N of n objects into a set X of x 'receivers'.
The 12-fold way relies on a fairly abstract approach of different function mappings from the set of objects N (balls) to the set of receivers X (bins).
The author indicates that this can be non-intuitive: "Counting modulo permutations of N or X (or both) is reflected by calling the balls or the boxes, respectively, "indistinguishable". This is an imprecise formulation (in practice individual balls and boxes can always be distinguished by their location, and one could not assign different balls to different boxes without distinguishing them), intended to indicate that different configurations are not to be counted separately if one can be transformed into the other by some interchange of balls or of boxes. This possibility of transformation is formalized by the action by permutations."
In other words, "identical" is not to be taken literally. 
It may mean that clearly distinct configurations of balls and boxes/bins are to be considered identical if we "don't care" about the differences.
A common example of "not caring" is with permutations. These configuration of balls into 2 boxes may be considered identical if you don't care about permutations: eg (1, 2) or (2, 1).
A more intuitive approach is the 20-fold way, described in the bottom half of the article.
It divides both objects (balls) and receivers (bins) into "distinct" and "identical" (or "indistinguishable") along with conditions which control how objects can be allocated to receivers.
Although this is at first sight easier to understand, the complexity remains because of the difficulty of deciding what "distinct" and "identical" actually mean.
Also, confusingly, some balls-and-bins structures that match the problem treat the objects as bins rather than balls as you might expect.
What I would suggest is to first pick problems in each of the standard categories and understand them thoroughly in their own right.
Let's take your two problems.
They are rather tricky ...
Notation: (n, x) represents the ways of choosing x items from a set of n items (n choose x).
Q1: There are (10,x) ways of choosing a committee of size x from 10 people.
This is a situation where permutations (arrangements) are not relevant.
A committee of (Fred, Wilma, Barney, Betty) is the same committee as (Betty, Barney, Wilma, Fred) - i.e. they are the same 4 people.
We know (10,x) = 210 => x = 4 since (10, 4) = 210.
The question does not mention roles in the committee so we must assume that a committee is defined just by the people in it (no roles).
The question says we need to "choose the same committee" with 12 people.
This can't mean the exact same set of committees we formed from the original 10 people, because there are always 210 of these no matter how many extra people we have to choose from.
The question must mean "how many committees of the same size can we form from 12 people"?
Thus, there are (12, 4) = 495 ways of choosing a committee of size 4 from 12 people.
Q2: We have to assume that a fruit salad is a mixture of k different fruits, where quantity does not matter.
In other words, for example, raspberry, blueberry and kiwifruit make one salad, regardless of how many raspberries/blueberries/kiwifruits you pick.
Let's say the bowl contains n types of fruit and we must choose k varieties of fruits to make each type of salad.
From these n fruits, we can create (n, k) different combinations of fruits.
The question is asking for (n, k).
Excluding one fruit from original bowl means n - 1 fruits, but we need to maintain same number of ingredients, k, so (n - 1, k) = 1330.
Reducing number of ingredients to k - 1 => (n - 1, k - 1) = 5985.
A standard formula for "combinations" is that (n, k) + (n, k + 1) = (n + 1, k + 1).
Thus, we can say (n - 1, k - 1) + (n - 1, k - 1 + 1) = (n, k)
So (n, k) = 1330 + 5985 = 7315.
We want (n, k) so the answer is 7315.
Note: The numerical solution by looking up table of binomial coefficients is (22, 4)
We want to express (n, k) or "n choose k" in terms of balls and bins.
To find the right balls-and-bins structure, ask yourself what does the solution look like?
A balls and bins system has 3 parts:
* Balls
* Bins
* Constraints on the bins
Approach (a) The first idea would be to label n balls with photos of each of the n people.
We then assign these 'people' balls to 2 'choice' bins: bin 1 being the chosen and bin 2 being the unchosen.
And we need constraints -- bin 1 must have exactly k balls and bin 2 must have exactly n - k balls.
The solution-set looks like a 'chosen' bin holding exactly k labelled balls (people) and a 'not chosen' bin holding exactly n - k labelled balls.
Approach (b) Consider the bins as people (so there are 10 bins) and interpret the balls as indicators (or flags) so that if a ball appears in a bin, it means that this bin is chosen.
There will be 4 balls representing 4 choices.
We need an additional condition so that the presence of a ball represents that bin being chosen.
The condition is that each bin can only have 1 ball or no balls.
A bin with 1 ball is an indicator that the bin (person) is chosen, a bin with no balls indicates the bin (person) is not chosen.
The solution-set looks like a row of labelled bins (people) with the 4 balls distributed between them to indicate the ways of choosing 4 of the people.
See also http://www.d.umn.edu/~jgreene/Combinatorics/Fall_2015/Notes_on_gen_fcns.pdf
In summary, questions to ask yourself are:


*

*Does "order" matter? Eg, is {Fred, Wilma} different to {Wilma, Fred}?

*Are the combinations distinguishable by "roles"?
If a committee has a chairperson, think of the chairperson wearing a "chairperson hat".
So {Fred with hat, Wilma} is different to {Fred, Wilma with hat}.

*In Q2, does a "fruit salad" (i.e. a combination of fruits) depend on the amount of each fruit?
In the real world, it does. But in this question, they say "different amounts of the same fruit do not make a difference".
So a fruit salad with 1 raspberry, 55 blueberries and 66 kiwifruit is considered the same as 99 raspberries, 7 blueberries and 3 kiwifruit.
That's why the fruits are "unlabelled" -- the quantity of the fruit does not matter, only the type of fruit.

*What does a balls-and-bins solution look like? See above.
