In how many ways can 12 cups be put in three trays in equal number ?

Question

Accepted Answer

The correct option is 2:
The problem requires finding the number of ways to partition 12 distinct items (cups) into three groups (trays) of equal size, which means $12 / 3 = 4$ cups per tray. In combinatorics, when dividing distinct items into equal-sized groups that are not explicitly ordered or labelled, the trays are considered identical or unordered. The calculation proceeds in two steps: first, find the number of ways to form the three groups, and second, divide by the number of ways those identical groups can be arranged. The number of ways to divide the 12 cups into three *ordered* groups of 4 is found using the multinomial coefficient: $\frac{12\!}{4\! \times 4\! \times 4\!}$, which calculates to $34,650$. Since the three groups are of the same size, their internal arrangement (the order of the trays) does not matter, and we must divide this result by $3\!$ (which is $3 \times 2 \times 1 = 6$) to account for this overcounting. Therefore, the total number of unique ways is $34,650 / 6 = 5,775$ ways.

The incorrect options are 1, 3, and 4:
The other options provided—36 ways, 5557 ways, and 4776 ways—are mathematically incorrect values for this specific combinatorial problem. Option 1, 36, is a severely underestimated value, which could result from a simple multiplication error or a misunderstanding of how combinations and factorials apply to distinct objects. The complexity of partitioning 12 distinct items into three sets of 4 necessitates a much larger number. Options 3 (5557) and 4 (4776) are not the precise result of the correct combinatorial formula. These options are typically included as distractors to test the student's calculation accuracy and their application of the specific rule for partitioning into equal-sized, unordered groups. The number of ways for this specific partition is a fixed mathematical quantity, and any result other than 5,775 indicates an error in the formula or the calculation.