Tips for understanding elementary combinatorics

In summary, the conversation between the speakers revolves around the topic of understanding combinatorics. One individual is struggling with the many variations of problems in this subject, while the other suggests breaking problems down into paradigm cases and using a specific procedure for solving novel problems. They also discuss the importance of exposure and practice in solving combinatorial problems.
  • #1
Magenta55
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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.
 
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  • #2
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.
 

1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting and arranging objects or events in a systematic way. It involves the study of combinations, permutations, and other methods of counting.

2. What are some real-world applications of combinatorics?

Combinatorics has various applications in fields such as computer science, statistics, genetics, and game theory. It is used in designing efficient computer algorithms, analyzing data sets, predicting genetic patterns, and solving mathematical puzzles.

3. What is the difference between permutations and combinations?

Permutations and combinations both involve selecting and arranging objects, but they differ in their restrictions. Permutations take into account the order of the objects, while combinations do not. In other words, the order matters in permutations, but not in combinations.

4. How can I apply combinatorics in probability problems?

Combinatorics plays a crucial role in probability problems by providing a systematic way to count the possible outcomes. It can help in calculating the total number of outcomes, the number of favorable outcomes, and ultimately, the probability of an event occurring.

5. What are some common pitfalls to avoid in combinatorics problems?

One common pitfall in combinatorics is forgetting to consider all possible cases or overcounting. It is essential to carefully define the problem and consider all constraints to avoid these errors. Another mistake is using the wrong formula for permutations or combinations, so it is crucial to understand the difference between the two.

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