Determine the number of length-$25$ sequences over $\{1,2,\ldots,6\}$ such that:
At least one element occurs $4$ times.
At least one element occurs $5$ times.
At least one element occurs $6$ times.
Determine the number of length-$30$ sequences over $\{1,2,\ldots,10\}$ such that:
At least two elements occur $4$ times each.
At least two elements occur $5$ times each.
At least one element occurs $6$ times.
I understand that counting the number of partitions is crucial. Here's my initial approach for the first one:
We need to partition our set of 25 positions within the string into four subsets: one subset of size $4$ (for the character digit appearing $4$ times), one subset of size 5 (for the character digit appearing 5 times), another subset of size 6 (for the character digit appearing $6$ times), and a final subset of size $10$ for the remaining characters. The partitioning can be accomplished in $\frac{25!}{4!5!6!10!}$ ways. Additionally, we must select the numbers to be repeated, which can be done in $\binom{6}{3}$ ways, and allocate them in $3!$ ways. Finally, we need to choose the remaining $10$ positions in $3^{10}$ ways. However, I suspect that my method counts duplicates due to the uniqueness of the $10$-element set. I would appreciate your assistance in resolving these concerns.