Permutations with Repetition (Multiset) Calculator
Permutations with Repetition (Multiset) Calculator
Enter the total number of items and the size of each group of identical items (like repeated letters), and this calculator counts the distinct arrangements.
When some of the items being arranged are identical to each other, swapping two of them around doesn't actually create a new, distinguishable arrangement — which means the plain factorial formula for permutations overcounts the true total, since it treats every item as unique even when several aren't. Dividing by the factorial of each repeated group's size corrects for this exactly: the formula n! divided by the product of each group's factorial is the classic approach for counting distinct arrangements of a word like "MISSISSIPPI," where several letters repeat multiple times.
- When some items are identical, swapping them doesn't create a new arrangement, so the plain n! formula overcounts — dividing by the factorial of each repeated group's size corrects for this.
- The formula is n! / (n₁! × n₂! × … × nₖ!), where n is the total count and each nᵢ is the size of one repeated group.
- This is exactly how anagram counting works: the word MISSISSIPPI has 11 letters (1 M, 4 I's, 4 S's, 2 P's), giving 11!/(1!4!4!2!) = 34,650 distinct letter arrangements.
How do I count arrangements when some items repeat?
Divide the factorial of the total number of items by the factorial of each group of identical items multiplied together.
How many distinct arrangements does MISSISSIPPI have?
11 letters total, with groups of 1 (M), 4 (I), 4 (S), and 2 (P): 11!/(1!×4!×4!×2!) = 39,916,800/1,152 = 34,650.
Permutations with Repetition (Multiset) Calculator
Enter the total number of items and the size of each group of identical items (like repeated letters), and this calculator counts the distinct arrangements.
When some of the items being arranged are identical to each other, swapping two of them around doesn't actually create a new, distinguishable arrangement — which means the plain factorial formula for permutations overcounts the true total, since it treats every item as unique even when several aren't. Dividing by the factorial of each repeated group's size corrects for this exactly: the formula n! divided by the product of each group's factorial is the classic approach for counting distinct arrangements of a word like "MISSISSIPPI," where several letters repeat multiple times.
- When some items are identical, swapping them doesn't create a new arrangement, so the plain n! formula overcounts — dividing by the factorial of each repeated group's size corrects for this.
- The formula is n! / (n₁! × n₂! × … × nₖ!), where n is the total count and each nᵢ is the size of one repeated group.
- This is exactly how anagram counting works: the word MISSISSIPPI has 11 letters (1 M, 4 I's, 4 S's, 2 P's), giving 11!/(1!4!4!2!) = 34,650 distinct letter arrangements.
How do I count arrangements when some items repeat?
Divide the factorial of the total number of items by the factorial of each group of identical items multiplied together.
How many distinct arrangements does MISSISSIPPI have?
11 letters total, with groups of 1 (M), 4 (I), 4 (S), and 2 (P): 11!/(1!×4!×4!×2!) = 39,916,800/1,152 = 34,650.