Permutations of Non-Distinct Objects

When objects are not distinct, the number of permutations is adjusted by dividing by the factorials of repeated items.

The formula is \( \frac{n!}{n_1!n_2!\cdots n_k!} \), where \( n_i \) are the counts of repeated objects.

Examples:

• The word LEVEL has \( \frac{5!}{2!2!} = 30 \) different permutations (since the letter L repeats twice and the letter E repeats twice.)
• The word MISSISSIPPI has \( \frac{11!}{4!4!2!} = 34650 \) permutations (since the letter I repeats $4$ times, the letter S repeats $4$ times, and the letter P repeats twice.)

This formula corrects for overcounting identical objects (since LEVEL = LEVEL).

In other words, whenever repetition occurs, we must adjust the usual \( n! \) formula by dividing by the number of identical objects.