Tips for understanding elementary combinatorics

[Magenta55]

Hi folks, can you guys share your experience and tips for understanding this subject?

I find the sheer amount of problems and their novelty very difficult to reconcile. I mean I understand the definitions and theorems well and can usually apply them in straight forward cases, but the many variations of problems is causing me trouble. For instance, we can have permutations, permutations of several cases, combinations without repetition, combinations with repetition, or even a mixture of these cases.

I have been fine with other topics that are not algorithmic in nature, for instance, it's easier for me to prove elementary topics in mathematics, even if the problems are novel. But when it comes to combinatorics, I can't solve a truly novel problem correctly the first time around. My problem is, I can only solve problems of similar types only after I have been exposed to how to solve them. Is this normal in this subject? Is the only remedy solving more problems?

Thanks.

 

[jambaugh]

With the diversity of problems you should break things down into paradigm cases and then, while reading a given example see to which case it matches up.
Examples: Drawing colored balls out of a bag i. with replacement, ii. without replacement. This is a standard paradigm problem and you can compare it to say a problem about forming committees out of various political parties in congress, or formulating meals out of various food groups, etc.

[edit] As to how to approach novel combinatorial problems, when it is an example of enumerating all cases, devise a procedure by which you would select a most specific case and consider the sequence of choices you must make. Count choices at each stage then multiply (if they are parallel choices).

Placing 5 cards from a deck in a row. You must choose the first card (52 choices) then the 2nd card (51 remaining choices)... and so on.