CAT 2026 DILR Distribution Sets: Coins, Chocolates and the Bounding Method for Allocation Problems
Treats allocation sets as a bounding problem. Teaches the four-step constraint-bounding method and the max-min TITA trick, with three solved sets — a clean maximum, a divisibility-driven range, and a capped minimum.
CAT 2026 DILR Distribution Sets: Coins, Chocolates and the Bounding Method for Allocation Problems
A distribution set rarely wants one exact answer. It hands you a fixed pile of coins or chocolates, a few rules about who gets more, and then asks for the largest or smallest amount a single person can hold. That is a bounding question, and treating it like a fill-in-the-blank puzzle is why aspirants burn time testing random splits. The clean approach is to bound each share between a minimum and a maximum, then push the variables to their limits. This guide gives you the constraint-bounding method for allocation problems in CAT 2026, with three solved sets and the max-min trick that TITA questions reward.
Why Distribution Sets Need a Different Toolkit
A distribution set starts with a fixed total of identical items and shares them among a few entities. The clues set relationships and limits: each person gets at least one, one share beats another, or two shares are tied by a ratio. Because the items are identical, position and order do not matter. Only the counts do.
This is what separates the family from seating or sequencing. There is no grid to fill and no order to fix, so diagram habits do not help. The real work is arithmetic under constraints, and the questions confirm it by asking for an extreme value rather than a single arrangement. Once you read a set as a bounding problem, the path opens up.
The Constraint-Bounding Method
The method is arithmetic and repeatable, which keeps you calm under the clock. Run these four steps in order and the extreme value falls out.
- Write a bounds table. Give every entity a minimum and a maximum from the explicit limits, such as at least 2 and at most 8.
- Translate every clue. Turn each relationship into an equation or inequality, then add the total as one more equation.
- Reduce to one driver. Substitute until a single variable controls the rest, and use divisibility to list only the valid integer values.
- Push to the extreme. To maximise one share, set the others to their minimums; to minimise it, set the others to their maximums, while every constraint still holds.
The fourth step is the one most aspirants skip. They find a split that works and submit it, but a working split is not the maximum or the minimum. The extreme value lives at the edge of the constraints, so you have to drive the other shares all the way to their limits before you read off the answer.
Distribution Sets vs Arrangement Sets
Labelling the family early decides your whole approach. The table below contrasts distribution with arrangement so you can spot the difference in your first read.
| Aspect | Distribution sets | Arrangement sets |
|---|---|---|
| What you find | Counts per entity | Positions per entity |
| Typical clue | A gets twice B | A sits next to B |
| Core move | Bound and push to the edge | Place and eliminate |
| Question style | Maximum or minimum share | Who sits where |
| Main tool | Bounds table and divisibility | Diagram or grid |
Turn Quiet Set Types Into Reliable Marks
Optima Learn drills the underrated DILR families, distribution included, so the sets others skip become the ones you bank on exam day.
Practise Allocation Sets3 Solved Distribution Sets
Here are three sets that climb in difficulty: a clean maximum, a divisibility-driven range, and a capped set with a minimum. Read the reasoning, then redo each one cold.
A library gives out 20 identical books among three departments X, Y and Z. Each department gets at least 2 books. X gets more than Y, and Y gets more than Z. What is the maximum number of books X can receive?
To maximise X, drive Y and Z to their smallest legal values. Z is at least 2, and Y must beat Z, so the smallest Y is 3. That leaves X equal to 20 minus 5.
X can be at most 15, and 15 still beats Y, so the bound is valid.
Thirty marbles are shared among four friends A, B, C and D, each getting at least one. A gets twice as many as B, and C gets three more than D. What is the maximum number A can have?
Write the total with B and D as drivers. Since A equals 2B and C equals D plus 3, the total becomes 2B plus B plus D plus 3 plus D, which simplifies cleanly.
For B to be a whole number, D must be a multiple of 3. The valid pairs are D = 3 with B = 7, D = 6 with B = 5, D = 9 with B = 3, and D = 12 with B = 1. A is largest when B is largest.
A is at most 14, when B = 7, D = 3 and C = 6, summing to 30.
Twenty-five students join four project groups, each group holding at least 3 and at most 8 students. Group 1 has more students than each of the other three groups. What is the minimum number of students in Group 1?
To make Group 1 as small as possible, make the others as large as they can be while staying below it. Each other group caps at 8, but they must also sit below Group 1. Test Group 1 at 7: the others can each be 6, giving 7 plus 6 plus 6 plus 6, which is 25.
Try Group 1 at 6 and the others can be at most 5 each, summing to 15, so the total cannot reach 25. So 7 is the floor.
The minimum for Group 1 is 7 students.
Common Traps in Allocation Problems
Most lost marks here come from a few repeatable slips. Watch for these as you build the bounds table.
- Dropping the lower bound. "Each gets at least one" is a real constraint. Forgetting it lets a share fall to zero and breaks the extreme value.
- Submitting a working split. A valid split is not the maximum. Push every other share to its limit before you read the answer.
- Ignoring divisibility. When a ratio links two shares, only some totals split into whole numbers. List the valid integer cases instead of assuming any value fits.
To make a single entity as large as possible, set every other entity to its smallest legal value first, then read off what remains. To make it as small as possible, do the opposite and fatten the others to their caps. The extreme always sits at the edge of the constraints, never in a comfortable middle split.
Scan the clues for three things: the fixed total, the lower bound on each share, and any ratio that links two entities. If a ratio appears, expect divisibility to limit the cases, and plan to list them. Naming these in ten seconds tells you whether the set resolves to one split or a range you must bound.
Distribution sets sit alongside the rest of the DILR families, so build them into your practice next to our guides on DILR mixed sets and DILR ranking sets. Slot allocation practice into your wider CAT preparation, and track your set-selection accuracy each week with the CAT preparation tracker.
The reward is a quiet, dependable scorer. A distribution set you can bound in two minutes becomes a calm pick during selection, the kind that protects your section score when flashier sets eat into your time. Keep the bounding method central to your CAT 2026 preparation and rehearse it until the bounds table writes itself as you read the clues.
Distribution Set Questions, Answered
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