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I've been studying stats, and currently taking my first ever engineering based stats course in college. It covers Probability extensively and other stats topics. Currently, I'm stuck on recognizing key points in a problem involving permutations / combinations vs. fundamental counting principle. I have 2 example problems and what would help the most is key things to look to recognize using the counting principle vs permutations / combinations formulas. Here's one that uses the permutations / combinations according to my student solutions manual

A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries.

  1. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?

  2. If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this?

  3. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?

  4. If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen?

  5. If 6 bottles are randomly selected, what is the probability that all of them are the same variety?

Here's one that uses counting principle

The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano).

a) How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto?

b) The manager of a radio station decides that on each successive evening (7 days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?

Any ideas would be helpful to recognize the clues in the problems.

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  • $\begingroup$ while the cases 2 to 5 are easy to understand, the 1st is not clear. Is it random too ? $\endgroup$
    – user354674
    Sep 12, 2016 at 22:27
  • $\begingroup$ The number of permutations is the product of the cardinalities of the sets that is drawn from. To get the combinations you use this formula: $\frac{\prod_{1 \leq i \leq r}\# X_i \colon \# X_i = n + r - i}{\prod_{1 \leq i \leq r} \# X_i \colon \# X_i = i}$ where r is the number of repetitions (r=1 for no repetitions I think. perhaps you only change the upper part of the fraction). If you have more equivalences you divide by them too. $\endgroup$
    – Emil
    Sep 13, 2016 at 5:14
  • $\begingroup$ The formula I wrote is wrong. Should be one variable for the number of chosen (in the product indices upper bound) and one variable for the number of repetitions. $\endgroup$
    – Emil
    Sep 13, 2016 at 5:22
  • $\begingroup$ (I think the equivalence you divide here is "sequences of different order but with the same elements are equivalent") $\endgroup$
    – Emil
    Sep 13, 2016 at 5:41

1 Answer 1

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It is not really a question of "versus".   They are often applied together.

In the first lot of problems, you are counting ways to select elements from sets (collections of distinct elements).   Sometimes you are also counting ways to arrange them.   That is combinations and permutations (respectively).

In the second lot of problems, you are performing selections from multiple sets, in sequence.   Thus each task can be divided into a series of independent sub-tasks; hence the Universal Principle of Counting is also used.

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