Factorial and Newton Symbol
This page introduces two important mathematical concepts: the factorial and the Newton symbol binomialcoefficient.
Factorial
The factorial of a positive integer n, denoted as n!, is defined as the product of all positive integers from 1 to n:
n! = 1 · 2 · 3 · ... · n
Example: 4! = 1 · 2 · 3 · 4 = 24
Highlight: By definition, 0! = 1 and 1! = 1
Vocabulary: Silnią 1 refers to the factorial of 1, which equals 1.
Newton Symbol (Binomial Coefficient)
The Newton symbol, also known as the binomial coefficient, is denoted as nk or nCk, where n and k are non-negative integers with k ≤ n.
Definition: The Newton symbol is defined as:
nk = n! / k!⋅(n−k!)
Vocabulary: Symbol Newtona is the Polish term for the Newton symbol.
Some important properties of the Newton symbol include:
- n0 = nn = 1
- nk = nn−k
- n+1k = nk + nk−1
Pascal's Triangle
Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.
Highlight: In Pascal's triangle, each number is the sum of the two numbers directly above it.
The first few rows of Pascal's triangle are:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Example: In the fourth row, 6 = 3 + 3, which corresponds to 32 = 21 + 22
Vocabulary: Trójkąt Pascala is the Polish term for Pascal's triangle.
The Newton symbol and Pascal's triangle have numerous applications in mathematics, including kombinatoryka combinatorics and probability theory. Understanding these concepts is crucial for solving problems involving Dwumian Newtona Newton′sbinomialtheorem and related topics.
