Permutations and Combinations Formula Sheet
All key Permutations and Combinations formulas for JEE Main & Advanced quick revision: counting principles, nPr, nCr, circular permutations, distribution, and derangements.
Every formula, key relation, and must-know result for Permutations and Combinations on one scannable page. Use it for last-minute JEE revision.
Fundamental Principle of Counting
| Principle | Operation | When to use | Result |
|---|---|---|---|
| Multiplication (AND) | $\times$ | Sequential / simultaneous independent events | $m \times n \times p \times \ldots$ |
| Addition (OR) | $+$ | Mutually exclusive choices | $m + n + p + \ldots$ |
“AND then” / “followed by” means multiply. “OR” / “alternatively” (mutually exclusive) means add. Most JEE problems combine both, e.g. (3 + 4) × 2.
Factorial
$$\boxed{n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1}$$| Quantity | Value / Property | Notes |
|---|---|---|
| $0!$ | $1$ | By definition; crucial for formulas |
| $1!$ | $1$ | — |
| Recursive | $n! = n \times (n-1)!$ | Also $(n+1)! = (n+1) \times n!$ |
Quick values: $2! = 2$, $3! = 6$, $4! = 24$, $5! = 120$, $6! = 720$, $7! = 5{,}040$, $10! = 3{,}628{,}800$.
Simplify ratios by cancelling: $\dfrac{10!}{7!} = 10 \times 9 \times 8 = 720$. Do not compute the full factorials.
Permutations (order matters)
Core formulas
$$\boxed{^nP_r = \frac{n!}{(n-r)!} = n \times (n-1) \times \ldots \times (n-r+1)}$$| Quantity | Formula | Notes |
|---|---|---|
| Arrange all $n$ distinct objects | $n!$ | $^nP_n = n!$ |
| Arrange $r$ from $n$ distinct | $^nP_r = \dfrac{n!}{(n-r)!}$ | Order matters |
| $^nP_0$ | $1$ | Arrange nothing |
| $^nP_1$ | $n$ | — |
| $^nP_r$ when $r > n$ | $0$ | Cannot select more than available |
| With repetition allowed | $n^r$ | $r$ positions, $n$ choices each |
Properties
$$\boxed{^nP_r = n \times {}^{n-1}P_{r-1}} \qquad \boxed{\frac{^nP_r}{^nP_{r-1}} = n - r + 1}$$$$\boxed{^nP_r = {}^nC_r \times r!}$$Permutations with Repeated (Identical) Objects
For $n$ objects where $p$ are alike of one kind, $q$ alike of another, $r$ alike of another, …:
$$\boxed{\frac{n!}{p! \cdot q! \cdot r! \cdot \ldots}}$$| Special case | Result |
|---|---|
| All objects distinct ($p=q=\ldots=1$) | $n!$ |
| All $n$ objects identical | $1$ |
Include letters that appear once (divide by $1!$). MISSISSIPPI: M(1), I(4), S(4), P(2) gives $\dfrac{11!}{1!\,4!\,4!\,2!} = 34{,}650$. More repetition means fewer distinct arrangements.
Circular Permutations
| Type | Can flip? | Formula |
|---|---|---|
| Circular (round table, fixed orientation) | No | $(n-1)!$ |
| Necklace / garland (reflection allowed) | Yes | $\dfrac{(n-1)!}{2}$ |
Derivation: $n$ linear arrangements ($n!$) collapse to one under rotation, so $\dfrac{n!}{n} = (n-1)!$.
Necklaces, garlands and bracelets can be flipped, so divide by 2. Round tables, circular races and clocks have a fixed orientation, so do not divide by 2. For restrictions in a circle, bundle objects then apply $(n-1)!$ to the units (e.g. A, B together: $(n-2)! \times 2!$ for $n$ people total counted as units).
Combinations (order does not matter)
Core formula
$$\boxed{^nC_r = \binom{n}{r} = \frac{n!}{r!\,(n-r)!}} \qquad \boxed{^nC_r = \frac{^nP_r}{r!}}$$Properties
| Property | Statement |
|---|---|
| Symmetry | $^nC_r = {}^nC_{n-r}$ |
| Pascal’s rule | $^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$ |
| Sum of all | $^nC_0 + {}^nC_1 + \ldots + {}^nC_n = 2^n$ |
| End values | $^nC_0 = {}^nC_n = 1$ |
| Linear values | $^nC_1 = {}^nC_{n-1} = n$ |
| Quick $^nC_2$ | $\dfrac{n(n-1)}{2}$ |
| Quick $^nC_3$ | $\dfrac{n(n-1)(n-2)}{6}$ |
Maximum value of $^nC_r$: at $r = n/2$ if $n$ even; at $r = \frac{n-1}{2}$ and $\frac{n+1}{2}$ (equal) if $n$ odd.
“Committee / team / group / choose” means combination. “Arrange / order / rank / podium” means permutation. If $r > n/2$, compute $^nC_{n-r}$ instead — e.g. $^{50}C_{48} = {}^{50}C_2 = 1{,}225$.
Distribution and Division
Distribution of identical objects to distinct recipients (stars and bars)
| Constraint | Formula |
|---|---|
| Each gets $\ge 0$ (no restriction) | $^{n+r-1}C_{r-1} = {}^{n+r-1}C_n$ |
| Each gets $\ge 1$ | $^{n-1}C_{r-1}$ |
Equivalent to non-negative (or positive) integer solutions of $x_1 + x_2 + \ldots + x_r = n$.
Distribution of distinct objects to distinct recipients
| Constraint | Formula |
|---|---|
| No restriction | $r^n$ |
| Each gets $\ge 1$ (inclusion–exclusion) | $r^n - {}^rC_1 (r-1)^n + {}^rC_2 (r-2)^n - \ldots$ |
The “each $\ge 1$” count also equals $r! \cdot S(n,r)$, where $S(n,r)$ is the Stirling number of the second kind.
Division into groups
| Situation | Formula |
|---|---|
| Into distinct groups of sizes $n_1, n_2, \ldots, n_k$ (multinomial) | $\dfrac{n!}{n_1!\,n_2!\,\ldots\,n_k!}$ |
| Into $k$ identical groups of equal size $m$ ($n = km$) | $\dfrac{n!}{(m!)^k \cdot k!}$ |
Identical objects to distinct recipients uses stars and bars. Dividing distinct objects into identical groups of the same size divides the multinomial by $k!$ — but only for groups that share a size; groups of different sizes are already distinguishable.
Selection of non-consecutive items
| Arrangement | Select $r$ non-consecutive from $n$ |
|---|---|
| In a line | $^{n-r+1}C_r$ |
| In a circle ($n \ge 2r$) | $\dfrac{n}{n-r}\,{}^{n-r}C_r$ |
Geometry of Points and Polygons
| Quantity | Formula |
|---|---|
| Handshakes among $n$ people | $^nC_2 = \dfrac{n(n-1)}{2}$ |
| Lines from $n$ points (no 3 collinear) | $^nC_2$ |
| Triangles from $n$ points (no 3 collinear) | $^nC_3$ |
| Diagonals of an $n$-gon | $^nC_2 - n = \dfrac{n(n-3)}{2}$ |
When some points are collinear, subtract the lines or triangles those collinear points would otherwise form.
Sum of Numbers Formed
Sum of all $n$-digit numbers formed using digits $d_1, d_2, \ldots, d_n$ (without repetition):
$$\boxed{(n-1)! \times (d_1 + d_2 + \ldots + d_n) \times \underbrace{111\ldots1}_{n \text{ ones}}}$$Derangements (no element in its original position)
$$\boxed{D_n = n!\left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \ldots + \frac{(-1)^n}{n!}\right) = n!\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$$| Relation | Formula |
|---|---|
| Recursive | $D_n = (n-1)(D_{n-1} + D_{n-2})$ |
| Single-step recurrence | $D_n = nD_{n-1} + (-1)^n$ |
| Approximation ($n \ge 7$) | $D_n \approx \dfrac{n!}{e}$ |
| Derangement probability | $\dfrac{D_n}{n!} \approx \dfrac{1}{e} \approx 0.368$ |
| Exactly $k$ in correct position | $^nC_k \cdot D_{n-k}$ |
| Identity | $\sum_{k=0}^{n} {}^nC_k \cdot D_k = n!$ |
First values: $D_0 = 1$, $D_1 = 0$, $D_2 = 1$, $D_3 = 2$, $D_4 = 9$, $D_5 = 44$, $D_6 = 265$, $D_7 = 1{,}854$.
About 37% of random permutations have no fixed point, and this stays near $1/e$ for all $n \ge 7$. For small $n$ use the recursion $D_n = (n-1)(D_{n-1}+D_{n-2})$.
Master Comparison: nPr vs nCr
| Aspect | Permutation $^nP_r$ | Combination $^nC_r$ |
|---|---|---|
| Order | Matters | Does not matter |
| Formula | $\dfrac{n!}{(n-r)!}$ | $\dfrac{n!}{r!\,(n-r)!}$ |
| ABC vs BAC | Different | Same |
| Keywords | Arrange, rank, podium | Select, choose, committee |
| Relation | $^nP_r = {}^nC_r \times r!$ | $^nC_r = \dfrac{^nP_r}{r!}$ |
Quick Revision Map
graph TD
A[Counting] --> B[Fundamental Principle: AND multiply / OR add]
A --> C[Permutations: order matters]
A --> D[Combinations: order does not matter]
C --> C1["Distinct: nPr = n!/(n-r)!"]
C --> C2["Repetition: n!/p!q!r!"]
C --> C3["Circular: (n-1)! ; necklace (n-1)!/2"]
D --> D1["Select: nCr = n!/(r!(n-r)!)"]
D --> D2["Distribute: stars & bars, multinomial"]
D --> D3["Derangements: Dn = n! sum (-1)^i/i!"]Key Takeaways
- AND means multiply, OR means add; check for restrictions before counting.
- Order matters means $^nP_r$; order does not matter means $^nC_r$, related by $^nP_r = {}^nC_r \times r!$.
- Identical objects reduce arrangements: divide $n!$ by the factorials of repetitions.
- Circular: fix one object so $(n-1)!$; divide by 2 only for flippable necklaces and garlands.
- Stars and bars for identical objects to distinct boxes; multinomial for division into groups; divide by $k!$ for identical equal-sized groups.
- Derangement probability $\approx 1/e \approx 0.368$; memorize $D_2=1, D_3=2, D_4=9, D_5=44$.