Permutations with Repetitions: Repeated Items Formula

Permutations with repetitions cover two special cases: arranging n objects where some are identical (like the repeated letters of "BANANA"), and filling r positions when each choice can be reused (like a PIN that allows repeated digits). Both need adjustments to the plain permutation formula.

This matters because real counting problems are full of repeats — repeated letters in words, repeated digits in codes, repeated items in a shipment. Using plain ⁿPᵣ on such problems overcounts, sometimes massively.

Key Concepts

• Identical objects: objects that cannot be told apart, such as the three A's in BANANA.
• Overcounting: plain n! treats identical objects as different, counting each real arrangement many times.
• Repetition allowed: each position can reuse any choice, independent of other positions.
• Division correction: dividing by the factorials of repeat counts removes the overcounting.

Formulas

• Identical objects: arrangements of n objects with p₁ identical of one type, p₂ of another, and so on (p₁ + p₂ + … + pₖ = n) = n! / (p₁! p₂! … pₖ!).
• Repetition allowed: ways to fill r ordered positions from n types with reuse = n^r.
• Both formulas assume the positions or slots are distinguishable; only the objects may repeat.

Worked Example

How many distinct arrangements can be made from the letters of BANANA? There are 6 letters in total: A appears 3 times, N appears 2 times, and B appears once. Using the identical-objects formula: 6! / (3! 2! 1!) = 720 / (6 × 2 × 1) = 720 / 12 = 60. So there are 60 distinct arrangements. Check the logic: plain 6! = 720 counts every swap of the three A's as different, but the A's are identical, so we divide by 3! = 6, and similarly divide by 2! = 2 for the N's.

Common Misconceptions

• "Divide by the repeats only when items are identical." Correct instinct — but students also divide when items are merely similar; the formula applies strictly to indistinguishable objects.
• "n^r and n!/(p₁!…) are the same thing." Wrong — they solve different problems: n^r reuses types across positions (like digits in a PIN), while the division formula arranges a fixed multiset of items.
• "The repeat counts need not add to n." Wrong — p₁ + p₂ + … + pₖ must equal n, since every one of the n objects belongs to some type.

Real-World Applications

• Anagrams: counting distinct rearrangements of words with repeated letters.
• PINs and codes: counting sequences when digits may repeat.
• Inventory: arranging shipments containing identical boxes.
• Genetics: counting sequences built from a small alphabet of repeated symbols.

When objects repeat, plain factorials overcount: divide by the factorial of each repeat count, or use n^r when positions can reuse types freely. Spot which kind of repetition you face, and the right formula follows.