What Is Distinguishable Permutations?
Distinguishable permutations are the arrangements that look different from each other. When some items are identical, swapping them gives no new arrangement, so the ordinary n! overcounts.
The fix is to divide n! by the factorial of each group of repeated items: n! ÷ (n₁! × n₂! × … × nₖ!). The word MISSISSIPPI, for example, has 11 letters with groups of 4, 4 and 2, and 34,650 distinguishable arrangements.
Type a word and the calculator counts the letters for you, or enter the group sizes directly.
How to Use the Distinguishable Permutations Calculator
- Type a word, or switch to group sizes.
- Check the counts the calculator found for each repeated item.
- Read the number of different arrangements.
Distinguishable Permutations Formula
Distinguishable permutations
Distinguishable Permutations Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Total number of items | |
| n₁, n₂, … | Sizes of the groups of identical items | |
| n! | Factorial of n | All arrangements if every item were different |
| Result | Number of distinguishable arrangements | n! ÷ product of group factorials |
Distinguishable Permutations Worked Example
With the example values loaded in the calculator above, the steps are:
- There are n = 11 items in groups of sizes 1, 4, 4, 2.
- Distinguishable permutations = n! ÷ (1! × 4! × 4! × 2!).
- = 39,916,800 ÷ 1,152 = 34,650.
Result: Distinguishable permutations = 34,650.
Worked example: MISSISSIPPI
The word has 11 letters: M once, I four times, S four times and P twice.
The count is 11! ÷ (1! × 4! × 4! × 2!) = 39,916,800 ÷ 1,152 = 34,650 different arrangements.
Why we divide
If every letter were different there would be 11! orderings. Each real arrangement is then counted once for every way to reorder the identical letters within their group, which is 4! for the I and the S and 2! for the P. Dividing by those removes the overcount.
Where Distinguishable Permutations Is Used
Counting anagrams
Find how many different arrangements a word has.
Probability problems
Count outcomes with repeated items, such as colored beads in a row.
Combinatorics coursework
Check permutations of a multiset with the division shown.
Common Distinguishable Permutations Mistakes to Avoid
- Count each repeated item carefully.
- Letter case is ignored, so A and a count as the same letter.
Distinguishable Permutations Terms and Definitions
- Distinguishable
- Arrangements that look different from each other.
- Multiset
- A collection in which items may repeat.
Distinguishable Permutations FAQs
How do you find distinguishable permutations?
Divide n! by the product of the factorials of the counts of each repeated item.
How many arrangements does MISSISSIPPI have?
34,650.
What if no letters repeat?
The answer is n!, the ordinary number of permutations.
Does the order of the groups matter?
No. The product in the denominator is the same in any order.