Distributive Arrangements for GMAT | Stars and Bars Method

50-Word Summary

Distributive arrangements describe how items are shared among groups, considering distinctness and restrictions. Example: distributing 7 identical chocolates among 3 children uses the relationship (n + r − 1)C(r − 1). Substituting n = 7, r = 3 gives 9C3 ways.

 

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Overview

Distributive arrangements in combinatorics explore how objects can be divided among groups, with the method depending on whether the items are distinct or identical and whether restrictions are applied. This article, along with the accompanying video, introduces classic variations such as distributing distinct items across recipients using multiplication, dividing identical items through the “stars and bars” relationship, and adjusting counts when conditions like “no group left empty” are imposed. The focus is not on memorizing outcomes but on recognizing which approach fits a given situation. In the context of GMAT preparation, these problems are important because they test clarity of reasoning, attention to conditions, and the ability to apply core counting principles under timed pressure.

 

Distribution in Combinatorics

Distribution problems in combinatorics study how objects can be shared among recipients. The approach changes depending on whether the objects are distinct or identical, and whether restrictions apply.

When the Object are Distinct, Multiply

If the objects are distinct, each object can be assigned independently.

For example, suppose 3 different toys are to be given to 2 children.

Each toy can go to either child, so the total number of distributions is 2^3 = 8.

When the Object are Distinct, Use Stars and Bars Method

If the objects are identical, we use the “stars and bars” relationship:

Number of distributions = (n + r − 1) C (r − 1)

Here, n is the number of identical items and r is the number of recipients.

For example, distributing 5 identical pencils among 3 students gives (5 + 3 − 1)C(3 − 1) = 7C2 = 21 possible distributions.

This distinction forms the foundation of distribution-based questions.

An Explanatory Distribution Example

Question: In how many ways can 10 chocolates be distributed among 4 children if…

1. All the chocolates are different and any child can get any number of chocolates.
2. All the chocolates are different and no child can get all the chocolates.
3. All the chocolates are similar and any child can get any number of chocolates.
4. All the chocolates are similar and every child gets at least one chocolate.

Case 1: All items different, any distribution allowed

10 different chocolates are to be given to 4 children.

Each chocolate has 4 possible recipients.

So, the total arrangements are 4^10

Case 2: All items different, no child gets all

From 4^10 total cases, subtract the 4 possibilities where one child receives all 10.

Therefore, the total arrangements are 4¹⁰ – 4

Case 3: All items identical, any distribution allowed

Here, we treat the problem as dividing 10 chocolates into 4 parts.

n = 10, r =4

The general formula is (n+r−1)C(r−1)

Therefore, the total arrangements are 13C3

Case 4: All items identical, each child gets at least one

First, give 1 chocolate to each of the 4 children, leaving 6 to distribute.

n = 6, r = 4

(n + r −1)C(r−1) here, yield (6+4−1)C(4−1) = 9C3.

Therefore, the total arrangements are 9C3

The key is not to memorize but to see the logic: different items use multiplication, identical items use separators, and restrictions shift the count of objects before distribution.

Key Takeaways

Distributive arrangements analyze how items can be shared among groups, depending on whether objects are distinct or identical, and whether restrictions apply. Distinct objects are assigned independently, often using the multiplication principle. Identical objects require the “stars and bars” relationship: (n + r − 1) C (r − 1). Restrictions such as “no group left empty” are handled by adjusting totals before applying the relationship. Recognizing these distinctions avoids common errors of overcounting or undercounting. Regular practice with varied cases, ideally reinforced through timed sets and a high number of GMAT mocks, helps build clarity, speed, and confidence.

Parting Thought

Distributive arrangements teach us that how we divide resources often matters more than the resources themselves. In GMAT preparation, success is not about unlimited hours but about distributing time wisely among concepts, practice, and review. In the MBA applications voyage, the same principle applies: achievements, experiences, and goals must be balanced and presented thoughtfully. Life, too, is an exercise in distribution, of energy, priorities, and relationships. The harmony lies in knowing that every distribution is a choice. Thoughtful choices create clarity, balance, and purpose, guiding us toward both academic excellence and meaningful personal success.