🎓 Math Formulas
32 formulas in 10 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Sets · Permutations · Combinations · Binomial Theorem · Complex Numbers · Matrices · Determinants · Vectors · Functions · Relations & Functions
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
| n(A), n(B) | sizes of sets A and B |
|---|---|
| n(A ∩ B) | number common to both |
Where it is used: To count elements in the union of two overlapping sets — surveys, group counting without double-counting.
In simple words: Add the two group sizes, then subtract the overlap so shared members aren't counted twice.
Example: n(A)=20, n(B)=15, common 5 → 20+15−5 = 30
n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
| n(...) | sizes of the sets and overlaps |
|---|
Where it is used: To count elements in three overlapping sets — survey problems with three categories.
In simple words: Add the three groups, remove each double-counted overlap, then add back the triple overlap.
Example: Used in 'students who like tea/coffee/juice' problems
Subsets = 2ⁿ, Proper subsets = 2ⁿ − 1
| n | number of elements in the set |
|---|
Where it is used: To count how many subsets a set has.
In simple words: Each element is either in or out, giving 2 choices each, so 2ⁿ subsets in all.
Example: Set {a,b,c} has 2³ = 8 subsets
(A∪B)' = A'∩B', (A∩B)' = A'∪B'
| ' | complement (not in the set) |
|---|
Where it is used: To simplify complements of unions and intersections — logic and set theory.
In simple words: The 'not' of an OR becomes an AND of 'nots', and vice versa.
Example: Not(A or B) = (not A) and (not B)
ⁿPᵣ = n! ÷ (n − r)!
| n | total items |
|---|---|
| r | items chosen and arranged |
| ! | factorial |
Where it is used: To count ordered arrangements — passwords, seat orders, race finishes where order matters.
In simple words: Counts how many ways to arrange r things out of n when the order matters.
Example: ⁵P₂ = 5!/3! = 20
n! = n × (n−1) × (n−2) × … × 2 × 1
| n | a whole number |
|---|---|
| 0! | defined as 1 |
Where it is used: The building block of permutations and combinations — counting arrangements.
In simple words: Multiply all whole numbers from n down to 1. It counts how many ways to order n things.
Example: 5! = 5×4×3×2×1 = 120
Circular arrangements = (n − 1)!
| n | number of items in a circle |
|---|
Where it is used: To count seatings around a round table, where rotations are the same.
In simple words: In a circle there's no fixed start, so fix one person and arrange the rest: (n−1)!.
Example: 5 people round a table → 4! = 24 ways
Arrangements = n! ÷ (p! × q! × …)
| n | total items |
|---|---|
| p, q | counts of each repeated item |
Where it is used: To count arrangements of a word or set with repeated letters.
In simple words: Divide by the factorials of the repeats, since swapping identical items doesn't make a new order.
Example: LEVEL: 5!/(2!·2!) = 30 arrangements
ⁿCᵣ = n! ÷ [r!(n − r)!]
| n | total items |
|---|---|
| r | items chosen |
| ! | factorial |
Where it is used: To count selections where order does NOT matter — teams, committees, lottery picks.
In simple words: Counts how many ways to pick r things out of n when the order doesn't matter.
Example: ⁵C₂ = 5!/(2!·3!) = 10
ⁿPᵣ = ⁿCᵣ × r!
| ⁿPᵣ | arrangements |
|---|---|
| ⁿCᵣ | selections |
| r! | r factorial |
Where it is used: To convert between selections and arrangements of r items from n.
In simple words: Arrangements = selections times the number of ways to order the chosen r items.
Example: ⁵P₂ = ⁵C₂ × 2! = 10 × 2 = 20
ⁿCᵣ = ⁿCₙ₋ᵣ, ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ
| n, r | total and chosen |
|---|
Where it is used: Useful identities for combinations — choosing r is the same as leaving out (n−r).
In simple words: Picking r to keep is the same as picking (n−r) to drop; all the combinations add to 2ⁿ.
Example: ⁵C₂ = ⁵C₃ = 10
ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ
| n, r | total and chosen |
|---|
Where it is used: The rule that builds Pascal's triangle and the binomial coefficients.
In simple words: Each number in Pascal's triangle is the sum of the two above it.
Example: ⁴C₂ + ⁴C₁ = 6 + 4 = 10 = ⁵C₂
Tᵣ₊₁ = ⁿCᵣ · aⁿ⁻ʳ · bʳ
| n | the power |
|---|---|
| r | term position (from 0) |
| a, b | the two terms of (a+b)ⁿ |
Where it is used: To find any single term in the expansion of (a+b)ⁿ without expanding the whole thing.
In simple words: A shortcut to jump straight to the term you want when expanding a bracket raised to a power.
Example: (x+2)⁵, term with x³: ⁵C₂·x³·2² = 40x³
(a + b)ⁿ = Σ ⁿCᵣ · aⁿ⁻ʳ · bʳ
| n | the power |
|---|---|
| ⁿCᵣ | the binomial coefficients |
Where it is used: To expand a bracket raised to a whole-number power without multiplying it out step by step.
In simple words: A shortcut to expand (a+b) raised to any power using the counting numbers ⁿCᵣ.
Example: (a+b)² = a² + 2ab + b²
Terms in (a+b)ⁿ = n+1; middle term at (n/2 + 1) if n even
| n | the power |
|---|
Where it is used: To locate the middle term of a binomial expansion.
In simple words: An expansion of power n has n+1 terms; the middle one is often asked for.
Example: (a+b)⁶ has 7 terms; middle is the 4th
ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ
| n | the power |
|---|
Where it is used: To find the sum of all coefficients in a binomial expansion (put a=b=1).
In simple words: Setting both terms to 1 shows all the coefficients add to 2ⁿ.
Example: (1+1)³ = 8 = 1+3+3+1
|z| = √(a² + b²), z = a + bi
| a | real part |
|---|---|
| b | imaginary part |
| i | √(−1) |
Where it is used: To find the size (distance from origin) of a complex number — used in engineering and physics.
In simple words: Treat the real and imaginary parts like the sides of a right triangle; the modulus is the hypotenuse.
Example: z = 3 + 4i → |z| = √(9+16) = 5
i² = −1, i³ = −i, i⁴ = 1 (i = √−1)
| i | the imaginary unit √(−1) |
|---|
Where it is used: The basic rules for simplifying powers of i in complex-number arithmetic.
In simple words: i squared is −1, and higher powers cycle in a pattern of four.
Example: i⁴ = 1, so i⁵ = i
z = a + bi → z̄ = a − bi
| a | real part |
|---|---|
| b | imaginary part |
Where it is used: Used to divide complex numbers and to find modulus (z·z̄ = |z|²).
In simple words: The conjugate just flips the sign of the imaginary part.
Example: z = 3+4i → z̄ = 3−4i
|z₁ z₂| = |z₁| × |z₂|
| z₁, z₂ | two complex numbers |
|---|
Where it is used: To find the modulus of a product of complex numbers.
In simple words: The size of a product equals the product of the sizes.
Example: |z₁|=2, |z₂|=3 → |z₁z₂| = 6
Order = (rows) × (columns) = m × n
| m | number of rows |
|---|---|
| n | number of columns |
Where it is used: To describe the size of a matrix, which decides whether operations like multiplication are possible.
In simple words: Count the rows and columns; a matrix with m rows and n columns is 'm by n'.
Example: A matrix with 2 rows and 3 columns is order 2×3
A × I = I × A = A
| A | any square matrix |
|---|---|
| I | the identity matrix |
Where it is used: The identity matrix acts like the number 1 for matrices — used in inverses and solving systems.
In simple words: Just as multiplying a number by 1 leaves it unchanged, multiplying a matrix by I leaves it unchanged.
Example: Any 2×2 matrix times the 2×2 identity stays the same
|A| = ad − bc, A = [[a, b], [c, d]]
| a, b, c, d | the four entries of the matrix |
|---|
Where it is used: To solve linear equations, find inverses and check if a matrix is invertible.
In simple words: Multiply the main diagonal, subtract the other diagonal's product.
Example: [[1,2],[3,4]] → 1·4 − 2·3 = −2
Area = ½ |det of the coordinate matrix|
| det | the 3×3 determinant of the vertices |
|---|
Where it is used: To find a triangle's area from coordinates using a determinant (Class 12).
In simple words: A neat determinant of the three corner points gives twice the area.
Example: Vertices (0,0),(4,0),(0,3) → area 6
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
| a…i | the nine entries by rows |
|---|
Where it is used: To evaluate a 3×3 determinant by expansion along the first row.
In simple words: Expand along the top row, alternating signs, using 2×2 determinants.
Example: Used to solve 3-variable equations
|v| = √(x² + y² + z²)
| x, y, z | the vector's components |
|---|
Where it is used: To find the length of a vector in 2D or 3D — physics (force, velocity), computer graphics.
In simple words: Square each direction component, add them, and take the square root to get the arrow's length.
Example: v = (2,3,6) → |v| = √(4+9+36) = 7
a · b = |a||b| cos θ
| |a|, |b| | the vectors' magnitudes |
|---|---|
| θ | angle between them |
Where it is used: To find the angle between vectors or the work done by a force — physics and geometry.
In simple words: Multiplies two arrows in a way that depends on how much they point the same way.
Example: If a⊥b (θ=90°), a·b = 0
|a × b| = |a||b| sin θ
| |a|,|b| | the magnitudes |
|---|---|
| θ | angle between them |
Where it is used: To find the area of a parallelogram formed by two vectors, and directions in physics (torque, force).
In simple words: Multiplies two arrows in a way that is largest when they point at right angles.
Example: If a∥b (θ=0), a × b = 0
â = a⃗ ÷ |a⃗|
| a⃗ | the vector |
|---|---|
| |a⃗| | its magnitude |
Where it is used: To find a vector of length 1 in the same direction — directions in physics and graphics.
In simple words: Divide a vector by its own length to get a pure-direction arrow of length 1.
Example: a⃗=(3,4), |a⃗|=5 → â = (0.6, 0.8)
cos θ = (a⃗ · b⃗) ÷ (|a⃗| |b⃗|)
| a⃗ · b⃗ | the dot product |
|---|---|
| |a⃗|,|b⃗| | magnitudes |
Where it is used: To find the angle between two vectors — forces, directions, geometry.
In simple words: The dot product over the two lengths gives the cosine of the angle between them.
Example: Perpendicular vectors → cosθ = 0, θ = 90°
y = f(x)
| x | the input |
|---|---|
| f | the rule |
| y | the output |
Where it is used: The basic idea of a function — a rule that turns each input into exactly one output. Foundation of all higher maths.
In simple words: A function is a machine: put a number in, apply the rule, and get one number out.
Example: f(x) = 2x+1 → f(3) = 7
Relations from A to B = 2^(m·n); Functions = nᵐ
| m | elements in A |
|---|---|
| n | elements in B |
Where it is used: To count possible relations and functions between two sets.
In simple words: Counts how many ways elements of one set can be linked or mapped to another.
Example: |A|=2, |B|=3 → functions = 3² = 9