How is the number of unique permutations for a word with repeated letters calculated using the example RATTATA?

Conditions

• The word contains repeated letters.
• All positions in the permutation are filled.

Reasoning, step by step

1. Count the total number of letters in the word ($n=7$ for RATTATA).
2. Calculate the factorial of the total length: $7! = 5040$.
3. Identify groups of identical letters (three 'A's and three 'T's).
4. Divide the total factorial by the factorial of the count of each identical group ($3!$ for A's and $3!$ for T's).
5. Compute the final value: $\frac{5040}{6 \cdot 6} = 140$.

Example

For RATTATA, the script states: 'We divide by $3!$ for the "A"s and another $3!$ for the "T"s. The formula becomes $\frac{7!}{3! \cdot 3!}$, which simplifies to 140 unique permutations.'

Common misconceptions

• Adding the factorials of the repeats instead of multiplying them in the denominator.
• Forgetting to divide by the factorial for every group of repeated letters.
• Assuming that swapping identical letters creates a new unique arrangement.

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Connected concepts

Explore next

• How would the calculation change if there were two different pairs of repeated letters?
• What is the number of permutations for the practice problem ALOMOMOLA?

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