# Combinations with repetition: Why is this question solved using arrangements/permutations with repetitons?

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?

• 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. Oct 15, 2019 at 4:44
• 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)
– user704743
Oct 15, 2019 at 4:53
• Please read this tutorial on how to typeset mathematics on this site. Oct 15, 2019 at 7:17

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 $$\sum_{k=0}^5\binom{5}{k}=(1+1)^5=2^5.$$