Permutations and Combinations formulas for JEE Main & Advanced

24 formula entries for Permutations and Combinations, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Fundamental Principle of Counting (FPC)

m \times n

(multiplication),

m + n

(addition)

When to use. Use when counting total outcomes for multi-step processes or choices from distinct categories.

Trap. Confusing independent events with mutually exclusive ones; multiplication is for sequences, addition for alternatives.

Permutation

P (n, r) = \frac{n!}{(n - r)!}

When to use. Use when arranging distinct objects in a sequence, e.g., seating arrangements, word formations, rankings.

Trap. Forgetting that order matters in permutations; using combinations instead.

Combination

C (n, r) = (\binom{n}{r}) = \frac{n!}{r! (n - r)!}

When to use. Use when selecting a subset from a set without regard to order, e.g., choosing teams, lottery numbers.

Trap. Using permutations when order is irrelevant, e.g., selecting a committee.

Permutations with Repetition

\frac{n!}{n_{1}! n_{2}! \dots n_{k}!}

where

n_{1} + n_{2} + \dots + n_{k} = n

When to use. Use when arranging objects with repetitions, e.g., arranging letters of 'MISSISSIPPI'.

Trap. Forgetting to account for identical objects; overcounting arrangements.

Circular Permutations

(n - 1)!

for distinct objects;

\frac{(n - 1)!}{2}

for arrangements where clockwise and anticlockwise are same (e.g., necklace).

When to use. Use for seating around a round table, arranging beads in a necklace, or any cyclic arrangement.

Trap. Forgetting to divide by 2 for arrangements like necklaces where flipping gives same order.

Combinations with Repetition (Stars and Bars)

(\binom{n + r - 1}{r})

When to use. Use for distributing identical items into distinct boxes, or selecting items with repetition allowed.

Trap. Using this formula when order matters; use $n^{r}$ instead.

Counting with Constraints (Inclusion-Exclusion)

∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣

for two sets; extend for more.

When to use. Use when counting with 'at least one' or 'none' conditions, e.g., arrangements where no two vowels are together.

Trap. Forgetting to add back intersections when subtracting multiple conditions.

Gap Method for Arrangements with Restrictions

Number of gaps = (number of other objects) + 1

When to use. Use when arranging objects such that certain items are not adjacent, e.g., no two boys sit together.

Trap. Forgetting to consider gaps at ends; they are valid positions.

Block Method for 'Must Be Together'

Number of ways = (arrangements of block + others)

\times

(arrangements within block)

When to use. Use when certain objects must always be adjacent, e.g., vowels together in a word.

Trap. Forgetting to multiply by internal arrangements of the block.

Factorial Notation

n! = n \times (n - 1) \times \dots \times 1

When to use. Used as building block for all permutation and combination formulas.

Trap. Forgetting that $0! = 1$ , not 0.

Number of Diagonals in a Polygon

\frac{n (n - 3)}{2}

When to use. Use when counting diagonals in any convex polygon.

Trap. Forgetting to subtract sides; using $(\binom{n}{2})$ directly gives total segments.

Counting with 'At Least' or 'At Most'

Total ways =

\sum_{i = 0}^{k} ways with exactly i

When to use. Use for selection problems like 'at least 2 girls in a committee'.

Trap. Forgetting to subtract the case of zero when using complement for 'at least one'.

Multinomial Theorem for Counting

\frac{n!}{n_{1}! n_{2}! \dots n_{k}!}

where

\sum n_{i} = n

When to use. Use when distributing distinct items into distinct groups with fixed group sizes.

Trap. Using this for identical objects; use stars and bars instead.

Number of Ways to Form a Committee with a Chairperson

(\binom{n}{r}) \times r

(or

n \times (\binom{n - 1}{r - 1})

)

When to use. Use when a committee has a designated leader or head.

Trap. Forgetting to multiply by $r$ after selecting committee; chairperson is distinct.

Counting with Identical Objects (Distribution)

(\binom{n + r - 1}{r - 1})

for non-negative integers;

(\binom{n - 1}{r - 1})

for positive integers.

When to use. Use for distributing identical candies to children, or solving $x_{1} + . . . + x_{r} = n$ with integer solutions.

Trap. Using stars and bars for distinct objects; use $r^{n}$ instead.

Derangements (Subfactorial)

! n = n! \sum_{k = 0}^{n} \frac{(- 1)^{k}}{k!}

When to use. Use for problems like 'no one gets their own hat back'.

Trap. Using $n!$ directly; that counts all permutations including fixed points.

Counting with 'No Two Consecutive'

Number of gaps = (number of unrestricted objects) + 1

When to use. Use for arrangements where certain items cannot be adjacent, e.g., no two red balls together.

Trap. Forgetting that restricted objects may be identical; use combinations for placement.

Number of Ways to Choose with Repetition (Ordered)

n^{r}

When to use. Use for passwords, PIN codes, or any sequence where each position has same options.

Trap. Using $n^{r}$ when order does not matter; use combinations with repetition.

Circular Permutations with Identical Objects

\frac{(n - 1)!}{n_{1}! n_{2}! \dots n_{k}!}

valid only when one object is unique (fix it); otherwise use necklace/Burnside counting

When to use. Use for arranging beads in a necklace with repeated colors.

Trap. Forgetting to fix one object; overcounts by factor of $n$ .

Counting with 'At Least One' Using Complement

Total - ways with none

When to use. Use for 'at least one vowel' or 'at least one girl' in selections.

Trap. Forgetting to compute total correctly; include all possibilities.

Number of Ways to Partition a Set into Groups

\frac{n!}{n_{1}! n_{2}! \dots n_{k}!}

for labeled groups; divide by

k!

if unlabeled and equal sizes.

When to use. Use for dividing a team into subgroups of specific sizes.

Trap. Forgetting to divide by $k!$ when groups are unlabeled and of same size.

Counting with 'No Two Vowels Together'

Number of gaps = (number of consonants) + 1

When to use. Use for word arrangement problems where vowels must not be adjacent.

Trap. Forgetting that vowels are distinct; use permutations for arranging vowels in gaps.

Binomial Coefficient Properties

(\binom{n}{r}) = (\binom{n}{n - r})

(\binom{n}{r}) + (\binom{n}{r - 1}) = (\binom{n + 1}{r})

When to use. Use for simplifying combinatorial expressions and in binomial theorem.

Trap. Forgetting that $(\binom{n}{r})$ is defined only for $0 \leq r \leq n$ .

Counting with 'Exactly' Conditions

(\binom{type size}{k}) \times (\binom{others}{r - k})

When to use. Use for selections with specific counts, e.g., exactly 2 boys in a committee.

Trap. Forgetting to multiply by ways to choose the remaining objects.

Values to remember

0! = 1 Factorial Notation
1! = 1 Factorial Notation
!1 = 0 Derangements (Subfactorial)
!2 = 1 Derangements (Subfactorial)
!3 = 2 Derangements (Subfactorial)
!4 = 9 Derangements (Subfactorial)

Read the full chapter: Permutations and Combinations for JEE, with what JEE asks and the common traps.