Pascal's triangle

E26830

mathematical object number triangle

Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.

Jump to: Surface forms Statements Referenced by

Observed surface forms (3)

Surface form	Occurrences
Khayyam triangle	1
Traité du triangle arithmétique	1
Yang Hui's triangle	1

Statements (49)

Predicate	Object
instanceOf	mathematical object ⓘ number triangle ⓘ
boundaryCondition	C(n,0) = 1 for all n ≥ 0 ⓘ C(n,n) = 1 for all n ≥ 0 ⓘ
combinatorialMeaning	C(n,k) counts k-element subsets of an n-element set ⓘ C(n,k) counts number of paths in a grid from (0,0) to (k,n-k) using unit steps ⓘ
constructionMethod	start with 1 at top and repeatedly add adjacent pairs to form next row ⓘ
containsOnly	nonnegative integers ⓘ
definingProperty	each entry is the sum of the two entries directly above it ⓘ
diagonalProperty	first diagonal consists of all 1s ⓘ fourth diagonal lists tetrahedral numbers ⓘ second diagonal lists natural numbers n ⓘ third diagonal lists triangular numbers ⓘ
entryFormula	C(n,k) = n choose k ⓘ
entryNotation	(n k) ⓘ C(n,k) ⓘ
field	algebra ⓘ combinatorics ⓘ probability theory ⓘ
generalization	multinomial theorem ⓘ surface form: Pascal's pyramid multinomial coefficients ⓘ
growthDirection	extends infinitely downward ⓘ
hasRow	row 0: 1 ⓘ row 1: 1 1 ⓘ row 2: 1 2 1 ⓘ row 3: 1 3 3 1 ⓘ row 4: 1 4 6 4 1 ⓘ
hasShape	triangular array ⓘ
historicalUseBeforePascal	China ⓘ India ⓘ Persia ⓘ
identity	sum_{k} C(n,k)^2 = C(2n,n) ⓘ ∑_{k} (-1)^k C(n,k) = 0 for n ≥ 1 ⓘ
knownAsInChina	Pascal's triangle self-linksurface differs ⓘ surface form: Yang Hui's triangle
knownAsInPersia	Pascal's triangle ⓘ surface form: Khayyam triangle
namedAfter	Blaise Pascal ⓘ
nthRowRepresents	coefficients of the binomial expansion of (x + y)^n ⓘ
parityPattern	mod 2 pattern forms Sierpiński triangle ⓘ
relatedTo	Bernoulli trials ⓘ binomial coefficients ⓘ binomial theorem ⓘ
rowIndexingStartsAt	0 ⓘ
rowSumProperty	sum of entries in nth row equals 2^n ⓘ
satisfiesRecurrence	C(n,k) = C(n-1,k-1) + C(n-1,k) ⓘ
symmetryProperty	C(n,k) = C(n,n-k) ⓘ
topElement	1 ⓘ
usedFor	computing binomial probabilities ⓘ counting combinations ⓘ expanding binomials ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Pascal's identity → involvesConcept → Pascal's triangle ⓘ

Pascal's triangle → knownAsInChina → Pascal's triangle self-linksurface differs ⓘ

this entity surface form: Yang Hui's triangle

Pascal's triangle → knownAsInPersia → Pascal's triangle ⓘ

this entity surface form: Khayyam triangle

Blaise Pascal → knownFor → Pascal's triangle ⓘ

Blaise Pascal → notableIdea → Pascal's triangle ⓘ

Blaise Pascal → notableWork → Pascal's triangle ⓘ

this entity surface form: Traité du triangle arithmétique

binomial theorem → relatesTo → Pascal's triangle ⓘ