Permutation, Combination & Probability

medium importance~1 Q in Tier 130 formulas⚡ 13 shortcuts6 subtopics

Overview Learn (6)Formulas (30)Examples (5)Practice (74)

Selections, arrangements and chance — the modern-maths block shared by SSC, Banking and CAT papers. The topic is small, but a handful of repeating patterns (word arrangements, committees, bag draws, two-dice sums) make its marks near-guaranteed with a little practice.

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Learn — 6 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

5 exam-level questions worked step by step.

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Practice & test

74 questions — untimed practice or a timed test with analysis.

Track record in the exam

Test difficulty mix (74 questions)

27 easy35 medium12 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

Forming numbers from given digits

very common

Some digits are given, and numbers of a fixed length must be formed — often 'without repetition' or with a condition like 'even' or 'greater than'.

Fill the restricted slot first (first digit or units place).

Multiply the slot choices.

Example

How many 3-digit numbers can be formed from 1, 2, 3, 4, 5 without repeating a digit?

Slots: $5 \times 4 \times 3$ = 60

Learn this in “Counting Principles: Product & Sum Rules” →

Arrange the letters of a word

very common

'In how many ways can the letters of WORD be arranged?' A repeated letter means an extra division.

All different: the answer is $n!$ .

Any repeat: divide by the factorial of each repeat count.

Example

In how many ways can the letters of APPLE be arranged?

5 letters, P twice: $\dfrac{5!}{2!}$ = 60

Learn this in “Permutations: Arrangements” →

Two people always (or never) together

common

'A and B always sit together' or 'no two girls sit together' in a row of people.

Together: make one block, arrange units and then inside the block.

Never together: arrange the rest, then place people in the gaps.

Example

In how many ways can 6 friends sit in a row with two particular friends always together?

Block + 4 others: $5! \times 2!$ = 240

Learn this in “Permutations: Arrangements” →

Committee of men and women

very common

A team must be chosen from two groups with a fixed number from each.

Use nCr for each group.

'And' multiplies the group counts; 'or' adds them.

Example

From 6 men and 4 women, a committee of 3 men and 2 women is formed. How many committees?

$^{6}C_3 \times ^{4}C_2 = 20 \times 6$ = 120

Learn this in “Combinations: Selections” →

Handshakes and league matches

common

Every pair shakes hands or plays exactly once — or the diagonals of a polygon are asked.

Pairs are $^{n}C_2 = \frac{n(n-1)}{2}$ .

Knockout matches are $n - 1$ , not $^{n}C_2$ .

Example

12 teams play a league where every pair meets once. How many matches are played?

$^{12}C_2 = \frac{12 \times 11}{2}$ = 66

Learn this in “Combinations: Selections” →

At least one in the selection

common

'At least one woman / defective / red ball' inside a selection question.

Total selections minus selections with none.

Example

A box has 10 bulbs, 4 of them defective. 2 bulbs are drawn. How many draws contain at least one defective bulb?

$^{10}C_2 - ^{6}C_2 = 45 - 15$ = 30

Learn this in “Combinations: Selections” →

Balls drawn from a bag

very common

A bag with coloured balls; one or two balls are drawn and a colour probability is asked.

One draw: favourable over total.

Two draws: multiply with shrinking totals, or count pairs with nCr.

Example

A bag has 4 red and 5 black balls. Two are drawn without replacement. Find P(one red, one black).

$\dfrac{4 \times 5}{^{9}C_2} = \dfrac{20}{36}$ = 5/9

Learn this in “Probability: Core Rules” →

Two dice and the sum

very common

'Two dice are thrown. Find P(sum is 7 / even / more than 9)'.

36 outcomes; count the ordered pairs that make the sum.

Example

Two dice are thrown. Find P(sum = 10).

Pairs: $(4,6), (5,5), (6,4)$ → $\dfrac{3}{36}$ = 1/12

Learn this in “Dice, Cards & Coins” →

Cards from a pack

common

'One/two cards are drawn from a pack of 52'; suits, colours, aces or face cards are asked.

Recall the deck facts: 13 per suit, 12 faces, 4 aces.

Two cards together: nCr pairs over $^{52}C_2 = 1326$ .

Example

Two cards are drawn from a pack. Find P(both hearts).

$\dfrac{^{13}C_2}{^{52}C_2} = \dfrac{78}{1326}$ = 1/17

Learn this in “Dice, Cards & Coins” →

Sharing identical items

common

'Distribute n identical items among r people', often with 'each gets at least one'.

Each at least one: $^{n-1}C_{r-1}$ .

Zeros allowed: $^{n+r-1}C_{r-1}$ .

Example

9 identical toffees are shared by 3 children, each getting at least one. In how many ways?

$^{8}C_2$ = 28

Learn this in “Distribution & Grouping” →

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