Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Permutations and Binomial Coefficients

In course

MATH-232: Probability and statistics (for IC)

Cours d'introduction à la théorie des probabilités et aux statistiques. ...

Description

This lecture covers the concepts of permutations and binomial coefficients, explaining how to calculate the number of different permutations of objects and types. It also delves into examples involving selecting committees and roles. The lecture further discusses multinomial coefficients and their applications in various scenarios.

Instructors (2)

Raphaël Jean Berthier

Emmanuel Abbé

Official source

Related concepts (32)

Binomial coefficient

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula which using factorial notation can be compactly expressed as For example, the fourth power of 1 + x is and the binomial coefficient is the coefficient of the x2 term.

Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.

Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.

Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example, for n = 4, The coefficient a in the term of axbyc is known as the binomial coefficient or (the two have the same value).

Gaussian binomial coefficient

In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as or , is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian .

Related lectures (49)

Combinatorial Mathematics

Explores combinatorial mathematics, covering permutations, combinations, and binomial coefficients, along with probability and statistics concepts.

Probability and Statistics

Introduces probability, statistics, distributions, inference, likelihood, and combinatorics for studying random events and network modeling.

Fundamental Limits of Gradient-Based Learning

Delves into the fundamental limits of gradient-based learning on neural networks, covering topics such as binomial theorem, exponential series, and moment-generating functions.

Continuous Random Variables: Basic Ideas

Explores continuous random variables and their properties, including support and cumulative distribution functions.