Your FREE GRE Math Flashcards 2026 – 200+ Cards

GRE Math Flashcards

Power rule (derivative)

$\dfrac{d}{dx} x^n = n x^{n-1}$ for any real exponent $n$.

Product rule

$(fg)' = f'g + fg'$.

Quotient rule

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$.

Chain rule

If $y = f(g(x))$ then $\dfrac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

Derivative of eˣ

$\dfrac{d}{dx} e^x = e^x$; and $\dfrac{d}{dx} a^x = a^x \ln a$.

Derivative of ln x

$\dfrac{d}{dx} \ln x = \dfrac{1}{x}$; and $\dfrac{d}{dx} \log_a x = \dfrac{1}{x \ln a}$.

Derivative of sin x

$\dfrac{d}{dx} \sin x = \cos x$.

Derivative of cos x

$\dfrac{d}{dx} \cos x = -\sin x$.

Derivative of tan x

$\dfrac{d}{dx} \tan x = \sec^2 x$.

Derivative of arctan x

$\dfrac{d}{dx} \arctan x = \dfrac{1}{1 + x^2}$.

Derivative of arcsin x

$\dfrac{d}{dx} \arcsin x = \dfrac{1}{\sqrt{1 - x^2}}$.

Implicit differentiation

Differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (so $y$ terms pick up a $\tfrac{dy}{dx}$), then solve for $\tfrac{dy}{dx}$.

Limit definition of the derivative

$f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$.

L'Hôpital's rule

For $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$: $\lim \dfrac{f}{g} = \lim \dfrac{f'}{g'}$, provided the right limit exists.

Squeeze theorem

If $g(x) \le f(x) \le h(x)$ near $a$ and $\lim_{x\to a} g = \lim_{x\to a} h = L$, then $\lim_{x\to a} f = L$.

Standard limit: sin x over x

$\lim_{x \to 0} \dfrac{\sin x}{x} = 1$.

Standard limit: (eˣ − 1)/x

$\lim_{x \to 0} \dfrac{e^x - 1}{x} = 1$.

Limit form of e

$\lim_{x \to \infty} \left(1 + \dfrac{a}{x}\right)^x = e^{a}$.

Continuity at a point

$f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$ (the limit exists and equals the function value).

Differentiable implies continuous

If $f$ is differentiable at $a$ it is continuous at $a$. The converse fails — e.g. $|x|$ at $0$.

Critical point

A point where $f'(x) = 0$ or $f'(x)$ is undefined; candidate for a local max, min, or inflection.

First-derivative test

At a critical point, $f'$ changing $+ \to -$ gives a local max; $- \to +$ gives a local min.

Second-derivative test

At $f'(c) = 0$: $f''(c) > 0$ means local min; $f''(c) < 0$ means local max; $f''(c) = 0$ is inconclusive.

Concavity & inflection point

$f'' > 0$ is concave up, $f'' < 0$ concave down; an inflection point is where concavity changes (often $f'' = 0$).

Mean Value Theorem

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, some $c$ gives $f'(c) = \dfrac{f(b) - f(a)}{b - a}$.

Rolle's theorem

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then $f'(c) = 0$ for some $c$ in $(a,b)$.

Extreme Value Theorem

A continuous function on a closed bounded interval $[a,b]$ attains an absolute maximum and minimum on that interval.

Intermediate Value Theorem

If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then $f(c) = N$ for some $c$ in $[a,b]$.

Related rates

Differentiate a relation between quantities with respect to time $t$, then substitute known rates to solve for the unknown rate.

Linear approximation

Near $a$: $f(x) \approx f(a) + f'(a)(x - a)$ — the tangent line approximates the function.

Fundamental Theorem of Calculus, Part 1

If $F(x) = \int_a^x f(t)\,dt$ then $F'(x) = f(x)$.

Fundamental Theorem of Calculus, Part 2

$\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.

Power rule for integrals

$\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \ne -1$; for $n = -1$, $\int \tfrac{1}{x}\,dx = \ln|x| + C$.

u-substitution

Let $u = g(x)$, $du = g'(x)\,dx$: $\int f(g(x)) g'(x)\,dx = \int f(u)\,du$ — the chain rule in reverse.

Integration by parts

$\int u\,dv = uv - \int v\,du$. Choose $u$ by LIATE (log, inverse trig, algebraic, trig, exponential).

Integral of eˣ

$\int e^x\,dx = e^x + C$.

Integral of 1/(1+x²)

$\int \dfrac{1}{1 + x^2}\,dx = \arctan x + C$.

Integral of sin x and cos x

$\int \sin x\,dx = -\cos x + C$; $\int \cos x\,dx = \sin x + C$.

Partial fractions

Decompose a rational function into simpler fractions whose denominators are the factors of the original denominator, then integrate term by term.

Area between two curves

$\displaystyle\int_a^b \big[ \text{top}(x) - \text{bottom}(x) \big]\,dx$, where top $\ge$ bottom on $[a,b]$.

Volume by disks

Revolving $y = f(x)$ about the x-axis: $V = \displaystyle\int_a^b \pi [f(x)]^2\,dx$.

Volume by cylindrical shells

About the y-axis: $V = \displaystyle\int_a^b 2\pi x\, f(x)\,dx$.

Arc length

$L = \displaystyle\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$.

Average value of a function

$f_{\text{avg}} = \dfrac{1}{b - a} \displaystyle\int_a^b f(x)\,dx$.

Improper integral

An integral with an infinite limit or an unbounded integrand; evaluate as a limit, e.g. $\int_1^{\infty} \tfrac{1}{x^2}\,dx = 1$ (converges).

Convergence of ∫ 1/xᵖ from 1 to ∞

$\int_1^{\infty} \dfrac{1}{x^p}\,dx$ converges iff $p > 1$.

Geometric series

$\displaystyle\sum_{n=0}^{\infty} a r^n = \dfrac{a}{1 - r}$ when $|r| < 1$; diverges when $|r| \ge 1$.

p-series test

$\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^p}$ converges iff $p > 1$. The harmonic series ($p = 1$) diverges.

nth-term (divergence) test

If $\lim_{n\to\infty} a_n \ne 0$, the series $\sum a_n$ diverges. (If the limit is 0, the test is inconclusive.)

Ratio test

Let $L = \lim |a_{n+1}/a_n|$: $L < 1$ converges, $L > 1$ diverges, $L = 1$ inconclusive. Best for factorials and exponentials.

Root test

Let $L = \lim \sqrt[n]{|a_n|}$: $L < 1$ converges, $L > 1$ diverges, $L = 1$ inconclusive. Best when $a_n$ has an nth power.

Integral test

For positive, decreasing $f$ with $a_n = f(n)$: $\sum a_n$ and $\int_1^{\infty} f\,dx$ both converge or both diverge.

Comparison test

If $0 \le a_n \le b_n$: $\sum b_n$ converges $\Rightarrow \sum a_n$ converges; $\sum a_n$ diverges $\Rightarrow \sum b_n$ diverges.

Limit comparison test

If $\lim \tfrac{a_n}{b_n} = c$ with $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ converge or diverge together.

Alternating series test

$\sum (-1)^n b_n$ converges if $b_n > 0$, $b_n$ is decreasing, and $b_n \to 0$.

Absolute vs conditional convergence

Absolute: $\sum |a_n|$ converges. Conditional: $\sum a_n$ converges but $\sum |a_n|$ diverges (e.g. the alternating harmonic series).

Taylor series

$f(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^n$. With $a = 0$ it is a Maclaurin series.

Maclaurin series of eˣ

$e^x = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \cdots$ (converges for all $x$).

Maclaurin series of sin x

$\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots$.

Maclaurin series of cos x

$\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots$.

Geometric (Maclaurin) series of 1/(1−x)

$\dfrac{1}{1 - x} = \displaystyle\sum_{n=0}^{\infty} x^n$ for $|x| < 1$.

Radius of convergence

The value $R$ such that a power series $\sum c_n (x-a)^n$ converges for $|x - a| < R$; often found by the ratio test.

Partial derivative

$\dfrac{\partial f}{\partial x}$ differentiates with respect to $x$, treating the other variables as constants.

Gradient vector

$\nabla f = \left( \dfrac{\partial f}{\partial x},\ \dfrac{\partial f}{\partial y} \right)$; it points in the direction of steepest increase of $f$.

Directional derivative

$D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}$, where $\mathbf{u}$ is a unit vector — the rate of change of $f$ in the direction $\mathbf{u}$.

Lagrange multipliers

To extremize $f$ subject to $g = c$, solve $\nabla f = \lambda \nabla g$ together with the constraint.

Second-partials (Hessian) test

With $D = f_{xx} f_{yy} - f_{xy}^2$: $D > 0, f_{xx} > 0$ min; $D > 0, f_{xx} < 0$ max; $D < 0$ saddle; $D = 0$ inconclusive.

Double integral over a rectangle

$\displaystyle\iint_R f\,dA = \int_c^d \!\! \int_a^b f(x,y)\,dx\,dy$ — an iterated integral (Fubini's theorem).

Green's theorem

$\displaystyle\oint_C (P\,dx + Q\,dy) = \iint_D \left( \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right) dA$.

Stokes' theorem

$\displaystyle\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$ — relates a line integral to the curl over the bounded surface.

Divergence theorem

$\displaystyle\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\,dV$ — flux through a closed surface equals the integral of the divergence.

Line integral of a scalar field

$\displaystyle\int_C f\,ds = \int_a^b f(\mathbf{r}(t))\,|\mathbf{r}'(t)|\,dt$.

Conservative vector field

$\mathbf{F} = \nabla \varphi$ for some potential $\varphi$; then line integrals are path-independent and $\nabla \times \mathbf{F} = \mathbf{0}$.

Separable differential equation

$\dfrac{dy}{dx} = g(x)h(y)$: separate to $\dfrac{dy}{h(y)} = g(x)\,dx$ and integrate both sides.

First-order linear ODE

$y' + P(x)y = Q(x)$: multiply by the integrating factor $\mu = e^{\int P\,dx}$, so $(\mu y)' = \mu Q$.

Exponential growth/decay ODE

$\dfrac{dy}{dt} = ky$ has solution $y = y_0 e^{kt}$ — growth if $k > 0$, decay if $k < 0$.

Second-order linear homogeneous ODE

$ay'' + by' + cy = 0$: solve the characteristic equation $ar^2 + br + c = 0$; roots give $e^{rx}$ solutions.

Newton's method

$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$ — iterates toward a root of $f$.

Curvature of a circle

A circle of radius $r$ has constant curvature $\kappa = \dfrac{1}{r}$.

Epsilon–delta definition of a limit

$\lim_{x\to a} f(x) = L$ means: for every $\varepsilon > 0$ there is $\delta > 0$ with $0 < |x - a| < \delta \Rightarrow |f(x) - L| < \varepsilon$.

Telescoping series

A series whose partial sums collapse, e.g. $\sum \left( \tfrac{1}{n} - \tfrac{1}{n+1} \right) = 1$.

Pythagorean trig identity

$\sin^2\theta + \cos^2\theta = 1$; dividing gives $1 + \tan^2\theta = \sec^2\theta$.

Double-angle formulas

$\sin 2\theta = 2\sin\theta\cos\theta$; $\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta$.

Euler's formula

$e^{i\theta} = \cos\theta + i\sin\theta$; so $e^{i\pi} + 1 = 0$.

Heron's formula

Triangle area $= \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \tfrac{a+b+c}{2}$ is the semi-perimeter.

Eigenvalue

A scalar $\lambda$ with $A\mathbf{v} = \lambda \mathbf{v}$ for some nonzero $\mathbf{v}$; found from $\det(A - \lambda I) = 0$.

Eigenvector

A nonzero vector $\mathbf{v}$ whose direction is unchanged (up to scaling) by $A$: $A\mathbf{v} = \lambda \mathbf{v}$.

Characteristic polynomial

$p(\lambda) = \det(A - \lambda I)$; its roots are the eigenvalues of $A$.

Trace

The sum of the diagonal entries of a square matrix; it equals the sum of the eigenvalues (with multiplicity).

Determinant of a 2×2 matrix

$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$. It equals the product of the eigenvalues.

Invertible matrix conditions

An $n \times n$ matrix is invertible iff $\det A \ne 0$ iff rank $= n$ iff its columns are linearly independent iff $0$ is not an eigenvalue.

Rank of a matrix

The dimension of the column space (= dimension of the row space) = number of pivots in row echelon form.

Rank–nullity theorem

For $T : V \to W$, $\operatorname{rank}(T) + \operatorname{nullity}(T) = \dim V$.

Kernel (null space)

The set of vectors $\mathbf{v}$ with $A\mathbf{v} = \mathbf{0}$; its dimension is the nullity.

Linear independence

Vectors are linearly independent if the only solution to $c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$ is all $c_i = 0$.

Basis

A linearly independent set that spans a vector space; every vector is a unique linear combination of basis vectors.

Dimension

The number of vectors in any basis of a vector space — an invariant of the space.

Diagonalizable matrix

$A = PDP^{-1}$ for a diagonal $D$; possible iff $A$ has $n$ linearly independent eigenvectors (guaranteed if all eigenvalues are distinct).

Vector space axioms

A set closed under vector addition and scalar multiplication satisfying associativity, commutativity, a zero vector, additive inverses, and the distributive/identity laws over a field.

Spectral theorem

A real symmetric matrix is orthogonally diagonalizable: it has real eigenvalues and an orthonormal basis of eigenvectors.

Orthogonal matrix

A square matrix with $Q^{\mathsf T} Q = I$ (columns orthonormal); it preserves lengths and $\det Q = \pm 1$.

Dot product

$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = \sum u_i v_i$; zero means the vectors are orthogonal.

Cross product

$\mathbf{u} \times \mathbf{v}$ is orthogonal to both, with magnitude $|\mathbf{u}||\mathbf{v}|\sin\theta$ (the area of the parallelogram they span).

Linear transformation

A map $T$ with $T(a\mathbf{u} + b\mathbf{v}) = aT(\mathbf{u}) + bT(\mathbf{v})$; every such map (between finite-dim spaces) is given by a matrix.

Image (range) of a linear map

The set of all outputs $T(\mathbf{v})$; its dimension is the rank, and it equals the column space of the matrix.

Cramer's rule

For $A\mathbf{x} = \mathbf{b}$ with $\det A \ne 0$: $x_i = \dfrac{\det A_i}{\det A}$, where $A_i$ replaces column $i$ with $\mathbf{b}$.

Group (definition)

A set with an associative operation, an identity element, and an inverse for each element. Abelian if the operation is also commutative.

Abelian group

A group whose operation is commutative: $ab = ba$ for all elements. Example: the integers under addition.

Order of an element

The smallest positive $n$ with $a^n = e$ (the identity); it divides the order of the group.

Order of a group

The number of elements in the group. By Lagrange's theorem, every subgroup's order divides it.

Lagrange's theorem

In a finite group $G$, the order of any subgroup $H$ divides $|G|$; the quotient $|G|/|H|$ is the index $[G:H]$.

Cyclic group

A group generated by a single element: $G = \langle a \rangle = \{ a^n \}$. Every group of prime order is cyclic.

Subgroup

A subset that is itself a group under the same operation — closed, contains the identity, and contains inverses.

Normal subgroup

$N \trianglelefteq G$ when $gN = Ng$ (i.e. $gNg^{-1} = N$) for all $g$; exactly the condition to form the quotient $G/N$.

Quotient group

$G/N$: the group of cosets of a normal subgroup $N$, with $(aN)(bN) = abN$. Its order is $|G|/|N|$.

Coset

For a subgroup $H$, a left coset is $gH = \{ gh : h \in H \}$. Cosets partition the group into equal-size blocks.

Group homomorphism

A map $\phi : G \to H$ with $\phi(ab) = \phi(a)\phi(b)$. An isomorphism is a bijective homomorphism.

Kernel of a homomorphism

$\ker\phi = \{ g : \phi(g) = e_H \}$ — always a normal subgroup of $G$; $\phi$ is injective iff the kernel is trivial.

First isomorphism theorem

For $\phi : G \to H$, $G/\ker\phi \cong \operatorname{im}\phi$.

Symmetric group Sₙ

The group of all permutations of $n$ elements; it has $n!$ elements and is non-abelian for $n \ge 3$.

Cyclic group Z mod n

$\mathbb{Z}/n\mathbb{Z}$: integers under addition mod $n$; the order of element $k$ is $n/\gcd(n,k)$.

Ring (definition)

A set with addition (an abelian group) and an associative multiplication that distributes over addition. With unity it has a multiplicative identity $1$.

Integral domain

A commutative ring with unity and no zero divisors: $ab = 0 \Rightarrow a = 0$ or $b = 0$.

Field

A commutative ring with unity in which every nonzero element has a multiplicative inverse (e.g. $\mathbb{Q}, \mathbb{R}, \mathbb{C}$).

Ideal

A subgroup $I$ of a ring under addition that absorbs multiplication: $rI \subseteq I$ for all $r$ in the ring.

Quadratic formula

For $ax^2 + bx + c = 0$: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Discriminant

$b^2 - 4ac$: positive gives two real roots, zero one (repeated) root, negative two complex-conjugate roots.

Vieta's formulas (quadratic)

For $ax^2 + bx + c = 0$: the roots sum to $-b/a$ and multiply to $c/a$.

Rational Root Theorem

Any rational root $p/q$ of an integer polynomial has $p \mid$ constant term and $q \mid$ leading coefficient.

Factor theorem

$(x - r)$ is a factor of $p(x)$ iff $p(r) = 0$ (i.e. $r$ is a root).

Logarithm rules

$\log(xy) = \log x + \log y$; $\log(x/y) = \log x - \log y$; $\log(x^k) = k\log x$.

Change of base formula

$\log_a x = \dfrac{\log_b x}{\log_b a}$.

Exponent rules

$x^m x^n = x^{m+n}$; $(x^m)^n = x^{mn}$; $x^{-n} = \dfrac{1}{x^n}$; $x^0 = 1$.

Euclidean algorithm

Find $\gcd(a,b)$ by repeated division: $\gcd(a,b) = \gcd(b,\, a \bmod b)$ until the remainder is 0.

Fermat's little theorem

If $p$ is prime and $p \nmid a$, then $a^{p-1} \equiv 1 \pmod{p}$; equivalently $a^p \equiv a \pmod{p}$.

Euler's theorem (number theory)

If $\gcd(a,n) = 1$, then $a^{\varphi(n)} \equiv 1 \pmod{n}$, where $\varphi$ is Euler's totient.

Euler's totient φ(n)

The count of integers in $\{1, \dots, n\}$ coprime to $n$. For a prime $p$, $\varphi(p) = p - 1$.

Chinese Remainder Theorem

If the moduli are pairwise coprime, a system $x \equiv a_i \pmod{n_i}$ has a unique solution mod $\prod n_i$.

Fundamental Theorem of Arithmetic

Every integer $> 1$ factors uniquely into primes (up to order).

Modular arithmetic

$a \equiv b \pmod{n}$ means $n \mid (a - b)$; arithmetic is done with remainders mod $n$.

Open set

A set in which every point has a neighborhood (an open ball) contained entirely in the set.

Closed set

A set whose complement is open; equivalently, it contains all of its limit points.

Closure of a set

The smallest closed set containing $A$; equals $A$ together with its limit points.

Compact set (Heine–Borel)

In $\mathbb{R}^n$, a set is compact iff it is closed and bounded; in general, every open cover has a finite subcover.

Connected space

A space that cannot be split into two disjoint nonempty open sets; intervals in $\mathbb{R}$ are connected.

Continuous image of a compact set

If $f$ is continuous and $K$ is compact, then $f(K)$ is compact (closed and bounded).

Continuous image of a connected set

The continuous image of a connected set is connected — the basis for the Intermediate Value Theorem.

Metric space

A set with a distance $d$ satisfying $d \ge 0$ (and $=0$ iff equal), symmetry, and the triangle inequality $d(x,z) \le d(x,y) + d(y,z)$.

Cauchy sequence

A sequence whose terms get arbitrarily close to each other; in a complete space (like $\mathbb{R}$) every Cauchy sequence converges.

Bolzano–Weierstrass theorem

Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

Supremum (least upper bound)

The smallest number $\ge$ every element of a set; the completeness of $\mathbb{R}$ guarantees it exists for any bounded-above set.

Uniform continuity

One $\delta$ works for all points at once: for every $\varepsilon$ there is $\delta$ with $|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon$.

Complex modulus

For $z = a + bi$, $|z| = \sqrt{a^2 + b^2}$ — its distance from the origin in the complex plane.

Complex conjugate

$\overline{a + bi} = a - bi$; note $z\bar z = |z|^2$.

Polar form of a complex number

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, where $r = |z|$ and $\theta = \arg z$.

De Moivre's theorem

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$.

nth roots of a complex number

$z = re^{i\theta}$ has $n$ roots $r^{1/n} e^{i(\theta + 2\pi k)/n}$, $k = 0,\dots,n-1$ — equally spaced on a circle.

Cauchy–Riemann equations

$f = u + iv$ is analytic where $\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}$ and $\dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}$.

Analytic (holomorphic) function

A complex function that is complex-differentiable on an open set; equivalently it satisfies the Cauchy–Riemann equations with continuous partials.

Cauchy's integral theorem

If $f$ is analytic on and inside a simple closed contour $C$, then $\displaystyle\oint_C f(z)\,dz = 0$.

Permutation P(n, k)

Ordered arrangements: $P(n,k) = \dfrac{n!}{(n-k)!}$.

Combination C(n, k)

Unordered choices: $\dbinom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$. Also $\binom{n}{k} = \binom{n}{n-k}$.

Binomial theorem

$(x + y)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}$.

Permutations with repeated items

Arrangements of $n$ items with repeats $n_1, n_2, \dots$: $\dfrac{n!}{n_1!\,n_2!\cdots}$.

Circular permutations

Seating $n$ distinct objects around a circle (rotations the same): $(n - 1)!$ arrangements.

Stars and bars

Nonnegative integer solutions of $x_1 + \cdots + x_k = n$: $\dbinom{n + k - 1}{k - 1}$.

Inclusion–exclusion

$|A \cup B| = |A| + |B| - |A \cap B|$; extends to more sets by alternating added and subtracted intersections.

Pigeonhole principle

If $n$ items go into $m < n$ boxes, at least one box holds $\ge \lceil n/m \rceil$ items.

Probability of an event

$P(E) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$ for equally likely outcomes; always between 0 and 1.

Addition rule of probability

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$; for mutually exclusive events the intersection is 0.

Independent events

$A$ and $B$ are independent iff $P(A \cap B) = P(A)\,P(B)$.

Conditional probability

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, for $P(B) > 0$.

Bayes' theorem

$P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}$.

Expected value

$E[X] = \sum x_i\, p_i$ (discrete) or $\int x f(x)\,dx$ (continuous) — the long-run average outcome.

Variance

$\operatorname{Var}(X) = E[(X - \mu)^2] = E[X^2] - (E[X])^2$; the standard deviation is $\sqrt{\operatorname{Var}(X)}$.

Linearity of expectation

$E[aX + bY] = aE[X] + bE[Y]$ — true even when $X$ and $Y$ are not independent.

Binomial distribution

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$; mean $np$, variance $np(1-p)$.

Poisson distribution

$P(X = k) = \dfrac{\lambda^k e^{-\lambda}}{k!}$; both mean and variance equal $\lambda$.

Uniform distribution on [a, b]

Density $\dfrac{1}{b-a}$; mean $\dfrac{a+b}{2}$, variance $\dfrac{(b-a)^2}{12}$.

Normal distribution

The bell curve $N(\mu, \sigma^2)$; about 68%, 95%, 99.7% of values fall within 1, 2, 3 standard deviations of the mean.

Central Limit Theorem

The sample mean of many independent identically distributed variables is approximately normal, regardless of the original distribution.

Mean, median, mode

Mean = average (sensitive to outliers); median = middle value (resistant); mode = most frequent value.

Triangle inequality

$|x + y| \le |x| + |y|$; in a metric space, $d(x, z) \le d(x, y) + d(y, z)$.

Cauchy–Schwarz inequality

$|\mathbf{u} \cdot \mathbf{v}| \le |\mathbf{u}|\,|\mathbf{v}|$, with equality iff the vectors are parallel.

Distance formula

Between $(x_1, y_1)$ and $(x_2, y_2)$: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Equation of a circle

$(x - h)^2 + (y - k)^2 = r^2$ — center $(h, k)$, radius $r$.

Slope of a line

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$; parallel lines share $m$, perpendicular slopes multiply to $-1$.

Sum of interior angles of a polygon

An $n$-sided polygon has interior angles summing to $(n - 2)\cdot 180^\circ$.

Volume of a sphere

$V = \dfrac{4}{3}\pi r^3$; surface area $4\pi r^2$.

Volume of a cylinder

$V = \pi r^2 h$; a cone is one-third of that, $\tfrac{1}{3}\pi r^2 h$.

Arithmetic sequence

Constant difference $d$: $a_n = a_1 + (n-1)d$; sum $S_n = \dfrac{n}{2}(a_1 + a_n)$.

Geometric sequence

Constant ratio $r$: $a_n = a_1 r^{\,n-1}$; finite sum $S_n = a_1\dfrac{1 - r^n}{1 - r}$.

Convergence of a sequence

$a_n \to L$ if for every $\varepsilon > 0$ there is $N$ with $|a_n - L| < \varepsilon$ for all $n > N$.

Set: union, intersection, complement

$A \cup B$ (in either), $A \cap B$ (in both), $A^c$ (not in $A$). De Morgan: $(A \cup B)^c = A^c \cap B^c$.

Cardinality / countability

A set is countable if it bijects with a subset of $\mathbb{N}$. $\mathbb{Q}$ is countable; $\mathbb{R}$ is uncountable.

Graph theory: handshake lemma

In any graph, the sum of all vertex degrees equals twice the number of edges.

Mathematical induction

Prove a base case, then assume $P(k)$ and prove $P(k+1)$; together these establish $P(n)$ for all $n \ge$ the base.

Law of total probability

For a partition $\{B_i\}$: $P(A) = \sum_i P(A \mid B_i)\,P(B_i)$.

Floor and ceiling functions

$\lfloor x \rfloor$ is the greatest integer $\le x$; $\lceil x \rceil$ is the least integer $\ge x$.

Numerical: trapezoidal rule

Approximates $\int_a^b f$ by trapezoids: $\dfrac{h}{2}\big[f_0 + 2f_1 + \cdots + 2f_{n-1} + f_n\big]$, with $h = \tfrac{b-a}{n}$.