How do you handle repeated letters when finding distinct arrangements?

Conditions

• The word contains repeated letters.
• We are finding the number of distinct permutations (arrangements).

Reasoning, step by step

1. Count the total number of letters in the word.
2. Identify the multiplicity (frequency) of each distinct letter.
3. Calculate the factorial of the total number of letters.
4. Calculate the factorial of the multiplicity for each repeated letter.
5. Divide the total factorial by the product of the individual multiplicity factorials.

Example

MISSISSIPPI has 11 letters with multiplicities 4, 4 and 2 for I, S and P. Its number of distinct arrangements is $11!/(4!4!2!)=34650$.

Common misconceptions

• Forgetting to divide by the factorials of the multiplicities.
• Dividing by the sum of the multiplicities instead of the product of their factorials.

Watch the explanation

Explore next

• Why do we divide by the factorials of the multiplicities?
• How do you calculate the number of combinations without repetition?
• What is the difference between permutations and combinations?