Multiset permutations (expand a^3 b^2 c^4): When a^3 b^2 c^4 is written out fully as a string of 9 letters (aaa bb cccc), how many distinct arrangements are possible?

Introduction / Context:
We seek the number of distinct permutations of a multiset where several letters repeat. The order matters, but identical letters are indistinguishable within each group.

Given Data / Assumptions:

• Total letters = 9 (3 a’s, 2 b’s, 4 c’s).
• Letters within the same species are indistinct.

Concept / Approach:
Use the multinomial formula: total permutations of n objects with groups of identical items is n! divided by the product of factorials of each group size.

Step-by-Step Solution:

Verification / Alternative check:
Sanity checks using smaller analogous sets (e.g., aabb) confirm the division by identical groups.

Why Other Options Are Wrong:
2520 doubles the correct value; 610 is not an integer factorization result from the required denominators.

Common Pitfalls:
Forgetting to divide by every identical-group factorial, or miscounting the total letters.

Final Answer:
1260