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In how many ways can we select five coins from a collection of $10$ consisting of one penny, one nickel, one dime, one quarter, one half-dollar, and five (identical) Susan B. Anthony dollars?

The answer is $2^5$, which does not make sense to me. Since this is a combinations with repetitions question, why does it use the arrangements formula: $n^r$. Shouldn't the answer be $\binom{14}{5}$, from using the combinations with repetitions formula?

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    $\begingroup$ As the dollars are identical, this is really a question about which non-dollar coins are chosen. Any subset of the five smaller coins can be chosen. $\endgroup$ Oct 15, 2019 at 4:44
  • $\begingroup$ true but since this question is taken from a chapter dealing with combinations with repetitions why is a different concept(permutations with repetition/arrangements) used instead of combinations with repetition C(n+r-1, r) $\endgroup$
    – user704743
    Oct 15, 2019 at 4:53
  • $\begingroup$ Please read this tutorial on how to typeset mathematics on this site. $\endgroup$ Oct 15, 2019 at 7:17

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The five coins can be $k$ coins taken from the set of one penny, one nickel, one dime, one quarter, one half-dollar ($5$ distinct elements) and $5-k$ identical Susan B. Anthony dollars for $k=0,1,2,3,4,5$. Therefore the number of ways to select them is $$\sum_{k=0}^5\binom{5}{k}=(1+1)^5=2^5.$$

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  • $\begingroup$ @PatrickPichart The person who asked can vote one or more answers and mark one answer as "accepted". Please visit math.stackexchange.com/tour $\endgroup$
    – Robert Z
    Nov 10, 2019 at 8:53

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