Factorial and Newton Symbol

This page introduces two important mathematical concepts: the factorial and the Newton symbol (binomial coefficient).

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 (n k) or nCk, where n and k are non-negative integers with k ≤ n.

Definition: The Newton symbol is defined as:

(n k) = n! / $k! · (n-k)!$

Vocabulary: Symbol Newtona is the Polish term for the Newton symbol.

Some important properties of the Newton symbol include:

1. (n 0) = (n n) = 1
2. (n k) = $n n-k$
3. $n+1 k$ = (n k) + $n k-1$

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 (3 2) = (2 1) + (2 2)

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's binomial theorem) and related topics.