GMAT Quant: Multiset Permutations | Repetitive Arrangements

50-Word Summary

Multiset permutations arise when certain elements are identical, reducing the total number of unique orderings. The relationship is expressed as n! ÷ (n₁! × n₂! × …). Example: arranging “BALLOON” gives 7! ÷ (2! × 2!) = 1,260 distinct arrangements, since L and O repeat.

 

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Overview

Multiset permutations or repetitive arrangements appear when some items are identical and others are distinct, making the counting process more subtle than simple permutations. Instead of treating every element as unique, we must account for repetitions to avoid overcounting. The explanatory video and the discussion that follow explore cases such as arranging colored balls of different counts and constructing words with repeated letters, each illustrating how the relationship between distinct and identical items alters the outcome. Building familiarity with such variations strengthens logical precision and enhances problem-solving efficiency. In your GMAT preparation, you must practice these adjustments carefully, as repetitive arrangements are often tested to check both attention to detail and conceptual depth.

 

Multiset Permutations on GMAT | Repetitive Arrangements

So far, we have considered cases where either every element is different or every element is identical.

In reality, many problems mix both.

To avoid overcounting, we use the relationship:

Unique arrangements = n! ÷ (n₁! × n₂! × …)

Here, n is the total number of items, and n₁, n₂, … are the counts of identical ones.

 A basic example

Take the word SUCCESS.

It has 7 letters, with S repeated 3 times and C repeated 2 times.

If all were distinct, there would be 7! arrangements.

Adjusting for repetitions:

7! ÷ (3! × 2!)

= 420 unique arrangements.

This simple example compactly illustrates how repetitive arrangements refine permutation counts.

Example 1: Colored Balls
Question: In how many ways can five red balls, four green balls, and three black balls be arranged in a row?

Here, you have twelve objects: five red balls, four green balls, and three black balls.

If all twelve were different, the number of arrangements would be 12!.

However, because identical items are indistinguishable, we must divide by the factorials of their counts.

The correct answer is: 12! / (5! × 4! × 3!)

This formula ensures that repeated arrangements caused by swapping identical objects are not overcounted.

Example 2: Words with Repeated Letters
Question: In how many ways can the letters of MISSISSIPPI be shuffled?

The word MISSISSIPPI has eleven letters in total.

If all were unique, we would have 11! arrangements.

But certain letters repeat: “I” occurs four times, “S” occurs four times, and “P” occurs twice.

The correct count of unique arrangements is: 11! / (4! × 4! × 2!)

Why This Matters

These repetitive arrangement problems illustrate how permutation principles adapt when items are not fully distinct. They test your attention to detail and your ability to balance simplicity with accuracy. The key is always to begin by assuming all are distinct, compute the factorial, and then adjust by dividing through the repeated groups.

Key Takeaways

Multiset permutations extend basic permutation ideas to situations where some items are identical. The principle is to begin with total arrangements, then adjust by dividing through the factorials of repeated elements, ensuring that identical swaps are not overcounted. Examples such as arranging colored balls or words with repeated letters highlight how precision in counting prevents inflated results. Mastery comes from recognizing the balance between distinct and indistinguishable objects. Consistent practice with varied cases, reinforced by attempting an adequate number of GMAT mocks, helps develop both accuracy and confidence, ensuring readiness for advanced combinatorics problems under timed conditions.

Parting Thought

Repetitive arrangements remind us that life is rarely about everything being unique; similarities often coexist with differences, and clarity comes from balancing both. In GMAT preparation, this means learning to adjust carefully, avoiding inflated counts and focusing on logic. The same perspective applies to the application process for a dream school such as ISB, where many candidates may share similar profiles, yet your unique qualities must stand out. Life itself mirrors this principle: while experiences may repeat, it is the thoughtful way we interpret them that creates originality and leads to lasting success.