DILR
Cubes
A one-page CAT 2026 reference for painted cube problems, covering the basic n-cube setup, corner/edge/face/interior counts, the reconciliation check, partially painted cubes, opposite versus adjacent…
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Selection
CAT’26 QUANT CHEATSHEET
Every combinations formula and shortcut you need for CAT 2026.
Selection is the quieter half of Permutation and Combination, and it is also the half that actually shows up more on CAT, hiding inside committee questions, group-division puzzles and geometry counts. The core idea is simple. Order does not matter, so nCr counts groups, not sequences. Where students lose marks is everywhere around that core idea, mixing up selection with arrangement, forgetting the extra division when groups end up equal in size, or grinding through cases that a quick complement would have closed in one line. This sheet gathers the moves you actually use, from the basic nCr formula and inclusion or exclusion rules, through identical items and group division, to diagonals, committees and complementary counting. Every box carries one worked example with real numbers, so you drill the method instead of just recognising the formula.
Every selection rule you need
1Basic Selection Formula
Choosing r items from n distinct items, with order not mattering.
nCr = n! / (r! × (n − r)!)
Example. Selecting 3 books from 8 gives 8C3 = 8! / (3! × 5!) = 56 ways.
CAT Hack. nCr always equals nC(n−r), so pick whichever side of the split has the smaller number.
2Selection vs Arrangement
Selection ignores order, arrangement cares which item goes where.
nPr = nCr × r!
Example. Choosing 3 of 5 friends for a trip is 5C3 = 10 ways, but seating those same 3 in a row is 5P3 = 10 × 3! = 60 ways.
Common Mistake. Reading ‘select’ as ‘arrange’ is the single most common P&C slip. Check whether roles or order exist.
3At Least One Item Selected
Total ways to pick any non-empty subset of n distinct items.
Non-empty subsets of n items = 2n − 1
Example. From 4 distinct toys, the number of ways to pick at least one is 24 − 1 = 15.
CAT Hack. Each item has 2 choices, in or out, giving 2ⁿ, then drop the one case where nothing is chosen.
4Fixed Inclusion
A committee that must always contain certain named people.
Must include k of n → choose the rest as (n−k)C(r−k)
Example. A team of 5 from 12 people must include 2 named seniors, so the count is 10C3 = 120, since only 3 more seats are open.
CAT Hack. Lock the required people in first, then treat the remaining seats as a smaller, plain selection.
5Fixed Exclusion
A committee that must never contain certain named people.
Must exclude k of n → choose from (n − k)C(r)
Example. A team of 5 from 12 people must exclude 2 named people, so the pool shrinks to 10C5 = 252.
Common Mistake. Do not subtract the excluded people from the answer. Remove them from the pool before you select.
6Selection with Restriction (Either/Or)
Exactly one of two specific people must be picked, never both.
Exactly one of A, B in the team = 2 × (n−2)C(r−1)
Example. A team of 4 from 10 needs exactly one of P or Q, giving 2 × 8C3 = 2 × 56 = 112 ways.
CAT Insight. The factor of 2 covers ‘P but not Q’ and ‘Q but not P’ as separate cases.
7Selecting from Identical Items
Choosing a quantity of items that are indistinguishable from each other.
Choosing r from n identical items has only 1 way
Example. Picking 4 pens from a box of 10 identical pens can be done in exactly 1 way, since the pens themselves are not distinct.
CAT Insight. Identical items kill the combinatorics. Only the count of chosen items matters, not which ones.
8Selecting from Mixed Identical Groups
Choosing items when several groups are internally identical.
At least 1 from each of k identical-type groups, sizes p₁…pₖ, gives (p₁+1)…(pₖ+1) − 1
Example. From 3 red, 2 blue, and 4 green identical balls, choosing at least one ball overall gives (3+1)(2+1)(4+1) − 1 = 60 − 1 = 59 ways.
CAT Hack. Each group contributes (count + 1) choices, including choosing zero from it, then remove the all-zero case.
9Dividing into Groups (Unequal, Unlabelled)
Splitting n distinct items into groups of fixed unequal sizes.
Split n into groups of sizes a, b, c = n! / (a! × b! × c!)
Example. Splitting 9 people into groups of 4, 3 and 2 gives 9! / (4! × 3! × 2!) = 1260 ways.
CAT Hack. No extra division by group factorial is needed here, because the group sizes already differ.
10Dividing into Equal Groups
Splitting n distinct items into groups of the same size.
Split n into k equal groups of size m = n! / ((m!)k × k!)
Example. Splitting 8 people into 2 groups of 4 gives 8! / (4!×4! × 2!) = 70 / 2 = 35 ways.
Common Mistake. Skipping the extra ÷ k! for equal groups is the classic error, since swapping two identical-size groups repeats the same split.
11Selecting Diagonals or Pairs from a Polygon
Counting line segments, diagonals, or pairs formed from n points.
Diagonals of an n-gon = nC2 − n
Example. A hexagon has 6C2 − 6 = 15 − 6 = 9 diagonals, after removing the 6 sides.
CAT Favourite. This diagonal count and its triangle-counting cousin, nC3, are recurring CAT favourites on polygons.
12Selecting a Committee with a Chairperson
Choosing a group and then marking one member as the leader.
Committee of r with 1 chair from n = nCr × r
Example. A team of 4 from 9 people with one captain gives 9C4 × 4 = 126 × 4 = 504 ways.
CAT Hack. Select the plain committee first, then multiply by r ways to pick the leader from inside it.
13Complementary Counting for Selection
Counting the easy opposite case and subtracting from the total.
Wanted = Total selections − Unwanted selections
Example. Choosing 3 of 10 people with at least 1 of 4 specific people equals total 10C3 minus none-of-them 6C3, so 120 − 20 = 100.
CAT Favourite. Whenever a question says ‘at least one’, try the complement first. It is almost always the shorter route.
Exam shortcuts, common traps and revision
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CAT Exam Shortcuts
- nCr = n! / (r! × (n − r)!), and nCr = nC(n − r)
- Selection ignores order, arrangement multiplies by r!
- At least one of n items = 2n − 1
- Must include k → select the rest as (n−k)C(r−k)
- Must exclude k → select from (n−k)C(r)
- Equal groups need an extra ÷ k! for the group swaps
- ‘At least one’ usually solves faster by complement
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Most Common CAT Traps
- Using nPr when the question only asks to select, not arrange
- Forgetting to divide by k! when the groups formed are equal in size
- Subtracting excluded people from the answer instead of from the pool
- Treating identical items as distinct and overcounting
- Missing the factor of 2 in an ‘exactly one of A or B’ condition
- Not trying the complement on an ‘at least one’ question
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30-Second Revision Box
- nCr = n! / (r!(n−r)!), symmetric in r and n−r
- Select first, arrange after, using nPr = nCr × r!
- Lock required people in, drop excluded people from the pool
- Equal-size groups need the extra ÷ k!
- At least one of n = 2n − 1
- Complement the ‘at least’ cases whenever you can
Selection questions reward pattern recognition more than raw calculation, so the fastest gains come from spotting which of these thirteen setups a question is actually built from. Read the question for order, restriction and group size before you touch a formula, and half the battle is already won. When you want an honest read on where you stand, run a full mock and check your projected percentile with the Optima Learn CAT score predictor, then work through the quant breakdowns on the blog. If you would rather build a plan with a person, book a free call with an Optima Learn mentor and shape one to your target.
