The second bin is forced to contain , leaving the other four letters in the first bin. Since order within a bin is irrelevant, this gives exactlyset partition of the specified bin sizes.
The first four letters may be permuted in ways and the last two in ways without changing this partition. The multiplication principle therefore gives
Choose the unordered pair of letters in the second bin; the first bin is then determined. The binomial coefficient givesThe first and second bins are distinguished, so their roles are not interchangeable.
The product isEvery permutation determines exactly one bin partition, and each such partition has exactly forty-eight internal orderings. Thus the fifteen classes partition the original sample space of arrangements, without overlap or omission.
For first-bin size , choose its contents in ways. Each choice arises from permutations, obtained by ordering the two bins internally. The seven ordered size choices giveHere . Each row expresses the same binomial coefficient identity . Even for the bins retain their first/second labels, so the number is twenty rather than ten; the braces denote bin sizes in the question's convention, not interchangeable bins.
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