These are the selections of comics: Action Comics, Superman, Captain Marvel, Archie, X-Man, and Nancy comics. How many ways are there to select six comics?

Question

BlackTom AI · Accepted Answer

Combinations with repetition (the stars-and-bars model) applies: we have 6 distinct comic titles and want to choose a total of 6 comics where repeats of a title are allowed and order does not matter.

1. Relevant concepts
- Selecting items from types with repetition allowed and order irrelevant is counted by the combinations-with-repetition formula: number of multisets of size r from n types = C(n + r − 1, r).
- Equivalently, the number of nonnegative integer solutions to x1 + x2 + ... + x6 = 6 (xi = number of copies of title i) equals C(6 + 6 − 1, 6) = C(11, 6).

2. Step-by-step reasoning
- Let the six titles be types 1..6; let xi denote how many copies of title i are chosen.
- We require x1 + x2 + x3 + x4 + x5 + x6 = 6 with each xi ≥ 0.
- The stars-and-bars theorem gives the count as C(6 + 6 − 1, 6) = C(11, 6).

3. Calculation details
C(11, 6) = 11! / (6! 5!)
Compute numerator product for the top 5 factors (since 11!/6! = 11·10·9·8·7):
11 · 10 · 9 · 8 · 7 = 55440
Denominator 5! = 120
55440 / 120 = 462

4. Why this is correct
- The mapping to nonnegative integer solutions is exact: each solution corresponds to a unique multiset selection of six comics (counts for each title). The combinations-with-repetition formula is the standard count for such multisets.

5. Why the other choices are incorrect
- If repeats were not allowed there would be C(6,6) = 1 way (choose each title once), not any of the listed alternatives.
- If order mattered (permutations) you'd get 6! = 720, or if you allowed ordered selections with repetition you'd get 6^6 = 46656—neither matches the given options.
- The numerical choices 370, 671, 500 do not arise from the correct combinatorial model here.

6. Conclusion
Therefore the number of ways to select six comics from six titles allowing repeats (order irrelevant) is 462, which matches the provided answer.

These are the selections of comics: Action Comics, Superman, Captain Marvel, Archie, X-Man, and Nancy comics. How many ways are there to select six comics?单项选择题

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