Permutations with Repetitions, how to select $n$ and $r$

Question

I'm doing Permutations and Combinations and I'm facing difficulty with Permutations with repetitions. My book states that if there are $n$ objects chosen $r$ at a time, the total number of permutations is $n^r$.

I cannot understand how to decide what $n$ is and what $r$

In the following problem:

In how many ways can 6 different rings be worn on 4 fingers?

While I can solve this using the Multiplication Principle, I keep thinking that $n$ is the number of rings (as in my textbook it says that there are '$n$' different things') and $r$ is the number of fingers.

@AndréNicolas Could you explain why this example is not a permutation with repetition? — N. F. Taussig, Sep 23 '15 at 22:56
Because the rings are all different. Note also that with the reasonable interpretation of different placement on fingers, the answer is not $4^6$. Fingers are not bins. — André Nicolas, Sep 23 '15 at 23:16

score 3 · Answer 1 · answered Sep 23 '15 at 23:30

It is reasonable to suppose that the order of rings on fingers matters. Blue then white then red on the ring finger looks better than red then blue then white.

To count the number of ways to choose the number of rings the various fingers will hold, use Stars and Bars. The number of choices is $\binom{9}{3}$. For each choice, the rings can be permuted in $6!$ ways, for a total of $\binom{9}{3}6!$.

N. F. Taussig · Accepted Answer · 2015-09-26T00:00:00.223

Definition. A permutation with repetition (or $r$-tuple or word) is an ordered selection of $r$ elements from a set of $n$ elements in which repetition is allowed.

By the Multiplication Principle, the number of words of length $r$ that can be formed with a set of $n$ elements is $n^r$ since we have $n$ choices for each of the $r$ times we make a choice.

Example. Consider the number of sequences of five letters we can form from the English alphabet if repetition is permitted. Since there are $26$ letters in the English alphabet, we have $26$ choices for each letter. This is our $n$. The number of times we choose a letter is $5$, which is our $r$. Thus, there are $26^5$ sequences of five letters of the English alphabet if repetition of letters is permitted.

Observe that $n$ is the number of choices, while $r$ represents the number of times you make that choice.

Andre Nicholas has pointed out in the comments and his solution that the number of ways that six different rings can be placed on four fingers is not a permutation with repetition (as I mistakenly thought) since the rings are not identical, so the order in which they are placed on the fingers matters.

Permutations with Repetitions, how to select $n$ and $r$

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