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3 days ago · Combinatorics is the mathematics of counting and arranging. Of course, most people know how to count, but combinatorics applies mathematical operations to count quantities that are much too large to be counted the conventional way. Combinatorics is especially useful in computer science.
- Multinomial Theorem
The multinomial theorem describes how to expand the power of...
- Knots
Knots are tangible and useful in everyday life, but they are...
- Hall's Marriage Theorem
Hall's marriage theorem is a result in combinatorics that...
- Linear Recurrence Relations
A linear recurrence relation is an equation that relates a...
- Closed-Form Expression
A closed-form expression is a mathematical process that can...
- Guarding a Museum
Given the layout of a museum, what is the minimum number of...
- Multinomial Theorem
3 days ago · In this lesson, we will learn how to use Pascal’s triangle to find the coefficients of the algebraic expansion of any binomial expression of the form (𝑎+𝑏)ⁿ.
2 days ago · Proof by strong induction. Step 1. Demonstrate the base case: This is where you verify that P (k_0) P (k0) is true. In most cases, k_0=1. k0 = 1. Step 2. Prove the inductive step: This is where you assume that all of P (k_0) P (k0), P (k_0+1), P (k_0+2), \ldots, P (k) P (k0 + 1),P (k0 +2),…,P (k) are true (our inductive hypothesis).
4 days ago · Blaise Pascal in 1654 proved Pascal's identity relating the sums of the p th powers of the first n positive integers for p = 0, 1, 2, ..., k. The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B 0, B 1, B 2,... which provide a uniform formula for all sums of powers.
2 days ago · If X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq). For example, imagine throwing n balls to a basket U X and taking the balls that hit and throwing them to another basket U Y .
1 day ago · This lift is what keeps the aircraft in the sky. Thrust, which is provided by the aircraft’s engines, is what makes the plane move forward through the air. When the air is pushed out the back of an aircraft’s engine, a reaction force occurs (thrust), and the airplane is pushed forward. When an airplane flies through the air, its shape ...
2 days ago · In the 1650s, Pascal and Huygens developed the concept of expectation on reasoning about the structure of games of chance. Pascal argued for equal division when chances are equal while Huygens extended the argument by considering strategies for a player who can make any bet with any opponent so long as its terms are equal. [7]
