Permutations and Combinations Questions for NDA

Fundamental Principle of Counting

If one task can be done in m ways and another independent task can be done in n ways, then both together can be done in m x n ways. This is the Multiplication Principle. If either task is done (not both), it is m + n ways — the Addition Principle. Think of it like a menu: if you pick one main AND one drink, multiply. If you pick one OR the other, add.

• Multiplication rule: AND means multiply
• Addition rule: OR means add
• Always identify whether tasks are done together or separately
• This principle is the foundation for all permutation and combination problems

Permutations — Arrangement Matters

A permutation is an arrangement of objects where ORDER MATTERS. Choosing 3 people from 5 for positions President, VP, Secretary is a permutation because who gets which post matters. The formula nPr gives the number of ways to arrange r objects chosen from n distinct objects. Think: Permutation = Position matters.

• nPr = n! / (n - r)!
• nPn = n! (all objects arranged)
• Permutation of n objects with repetition allowed: n^r
• Arrangements of n objects where p are identical of one type, q of another: n! / (p! x q!)
• Circular arrangement of n distinct objects: (n - 1)!
• Circular arrangement with necklace/bracelet (no clockwise/anticlockwise difference): (n - 1)! / 2

Combinations — Selection Matters, Order Does Not

A combination is a selection where ORDER DOES NOT MATTER. Choosing 3 students from 10 for a team is a combination because the team is the same regardless of the order you picked them. nCr is always smaller than or equal to nPr for the same values. Think: Combination = Committee (no ranks).

• nCr = n! / (r! x (n - r)!)
• nCr = nC(n-r) — choosing r is same as leaving out (n-r)
• nC0 = nCn = 1
• nC1 = n
• nCr + nC(r-1) = (n+1)Cr — Pascal's identity, useful for verification
• Sum of all combinations: nC0 + nC1 + ... + nCn = 2^n

Key Special Cases and Restrictions

NDA frequently tests problems with conditions like always include certain persons, never include certain persons, always together, or never together. The strategy is: fix the restricted objects first, then arrange the rest. For always together treat the group as a single unit. For never together use total minus unwanted cases.

• Always together: Tie them as one block, arrange block internally and externally
• Never together: Total arrangements minus arrangements where they ARE together
• At least one: Total minus none selected (use 2^n logic)
• Division into groups: Be careful whether groups are identical or distinct
• Selecting from multiple groups: Multiply individual combinations

Permutation vs Combination — Quick Decision

The single biggest skill in this topic is deciding which formula to use. Ask yourself: does the ORDER or ARRANGEMENT make a difference to the outcome? If yes, use permutation. If only the GROUP or SET matters, use combination. Keywords like arrange, order, rank, queue, sequence suggest permutation. Keywords like select, choose, committee, group, team suggest combination.

• Arrange, rank, sequence, queue = Permutation
• Select, choose, committee, group, team = Combination
• Permutation is always greater than or equal to combination for same n and r
• nPr = r! x nCr — permutation is just combination with internal arrangements
• When in doubt, ask: does swapping two items give a different outcome?

• Using nPr instead of nCr in selection problems — always check if order matters before choosing formula
• Forgetting to divide by repetition factorials in word arrangement problems like MISSISSIPPI or ARRANGE
• In circular arrangements, students use n! instead of (n-1)! — remember one person is fixed as reference
• In at-least-one problems, students forget to subtract the empty set case from 2^n, giving 2^n instead of 2^n - 1
• Confusing nCr symmetry: students recalculate nC7 when it equals nC(n-7), wasting time

• P for Position, C for Committee — if position matters use P, if only group matters use C
• CIRCULAR = fix one, arrange rest: think of one person glued to a chair, others move around (n-1)!
• REPEATED LETTERS = divide by the drama: each repeated letter creates identical clones so divide their factorials
• 2^n = total subsets including empty, 2^n - 1 = at least one selected. Think of a light switch for each item: on or off = 2 choices per item

• NDA typically asks 3 to 5 questions from this topic; word arrangement and committee selection are the most common question types in previous year papers
• Circular arrangement appears almost every year — remember (n-1)! for table seating and (n-1)!/2 for necklace problems
• Questions involving geometry (number of triangles, diagonals, lines from points) are based on combinations — diagonals of n-gon = nC2 - n
• Time-saving tip: use nCr = nC(n-r) to reduce calculation; computing 20C17 is far easier as 20C3 = 1140
• If a question says at least 1 or at most k, always use the complement method (total minus opposite condition) — it is always faster than adding individual cases