NDA Maths · Permutation & Combination

Combinations & Selections

Selecting a group where order doesn't matter — straight nCr selections, subsets, and the constrained selections (at-least / at-most, compulsory members, complementary counting).

All Permutation & Combination notes NDA Maths strategy

Why this matters

Selection problems hinge on spotting that order is irrelevant. The high-value trick is complementary counting — 'at least one' is total minus none — which beats summing cases.

Concept 1 of 2

Selections and subsets

Intuition

A combination counts groups, not orders. Choosing

r

from

n

^nC_r

; the number of subsets of an

n

-set is

2^n

. Fixing compulsory members just reduces the pool and the number to choose.

Definition

Choose $r$ of $n$ (order irrelevant): $^nC_r$ . Subsets of an $n$ -element set: $2^n$ . Compulsory members: if $k$ must be included, choose the rest: $^{n-k}C_{r-k}$ . Probabilities use $\dfrac{\text{favourable }C}{\text{total }C}$ .

Worked example

In how many ways can a committee of 3 be chosen from 8 people?

Order irrelevant: $^8C_3=\dfrac{8\cdot7\cdot6}{3!}$ .

Answer:

56

Practice this conceptself-check · 4 quick reps

Try it yourself

A team of 5 is chosen from 10 players, 2 of whom must be included. How many teams?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Choose $r$ of $n$ , order irrelevant?
2.
Number of subsets of an $n$ -set?
3.
$^8C_3=$ ?
4.
$k$ compulsory members of $r$ from $n$ ?

From the bank · past-year question

Example 1Permutation & CombinationEASY

In how many ways can a team of 5 players be selected from 8 players so as not to include a particular player?

[Q44 · Apr · 2021]

Drill 5 more on selections and subsets

Concept 2 of 2

Constrained selection: at-least, at-most, cases

Intuition

When a selection must satisfy 'at least one of X', it's almost always faster to count the complement (total minus the forbidden 'none') than to sum cases. Genuinely multi-part constraints split into disjoint cases that add.

Definition

At-least-one: $\text{total}-\text{none}$ . **At-most $k$ :** sum $^nC_0+\cdots+^nC_k$ . Cases: when the constraint forces distinct sub-situations (e.g. 'choose 3 from 4 women + 3 men with a balance rule'), count each disjoint case and add. Watch for over-/under-counting at the boundaries.

Worked example

From 6 programmers and 4 typists, choose 5 with at least one typist. How many ways?

Total $^{10}C_5=252$ ; none-typist (all programmers) $^6C_5=6$ .
At least one $=252-6$ .

Answer:

246

Practice this conceptself-check · 4 quick reps

Try it yourself

How many selections of at most 2 items from 5 distinct items?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
'At least one' is best counted as?
2.
'At most $k$ ' selections?
3.
At least one typist, 5 of 6+4, none $=^6C_5$ : answer?
4.
When do you sum cases?

From the bank · past-year question

Example 2Permutation & CombinationHARD

A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they invite 3 women and 3 men so that 3 of them are the man's relatives and 3 are his wife's relatives?

[Q5 · Apr · 2024]

Drill 4 more on constrained selection: at-least, at-most, cases

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Permutation & CombinationMODERATE

Committee of 3 from 4 men and 5 women. P(exactly 2 men)?

[Q119 · Apr · 2018]

Example 2Permutation & CombinationEASY

What is the number of positive integer solutions of

x + y + z = 5

[Q19 · Apr · 2025]

Example 3Permutation & CombinationEASY

There are 17 cricket players, out of which 5 players can bowl. In how many ways can a team of 11 players be selected so as to include 3 bowlers?

[Q16 · Sep · 2018]

Example 4Permutation & CombinationMODERATE

From 6 programmers and 4 typists, an office wants to recruit 5 people. What is the number of ways this can be done so as to recruit at least one typist?

[Q20 · Apr · 2019]

Example 5Permutation & CombinationMODERATE

A set

S

contains

(2n+1)

elements. There are 4096 subsets of

S

which contain at most

n

elements. What is

n

equal to?

[Q14 · Sep · 2023]

Drill every past-year question on this subtopic

11 questions from the bank — paginated, with cart and Word-export support.

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