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Permutations record order. Distinct n objects have n! complete arrangements; indistinguishable repeated groups require dividing out their internal exchanges. Specify the multiplicities, whether all objects are used and whether rotations count as different arrangements.
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From 546 to 590 seconds, labeled permutations collapse to the same arrangement of identical letters, so the proposed 4!3! mapping is not injective and the answer is rejected.
From 36 to 91 seconds, the repeated-letter example removes exchanges within identical groups: 7!/(3!3!)=140. This is finite arrangement counting, not a probability calculation.
A reviewed path from functions and finite counting to arrangements, selections, divisibility and the Euclidean algorithm. Every new concept is paired with verified video evidence, while the prerequisite arrows describe this map’s editorial learning order.
We divide by these factorials to correct for overcounting. When calculating , we treat all items as distinct.
Conditions: Calculating permutations of a multiset.; Some elements in the set are indistinguishable.
To calculate the unique permutations, first determine the total arrangements assuming all letters are distinct, which is for the 7-letter word RATTATA. Then, adjust for duplicates by dividing by the factorial of the count of each repeating letter group.
Conditions: The word contains repeated letters.; All positions in the permutation are filled.
The word ALOMOMOLA has 9 letters. It contains two 'A's, two 'L's, two 'M's, and three 'O's.
Conditions: Word length is 9.; Letter frequencies: , , , .