"In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial."
A formula for expanding the powers of a binomial. It also shows how to compute coefficients of terms in the expansion.
Combinations and Permutations: Understanding the difference between combinations and permutations, and how to calculate them.
Factorial Notation: Understanding factorial notation and how to use it to calculate the number of permutations and combinations.
Pascal's Triangle: Understanding Pascal's Triangle and how to use it to calculate the coefficients in the binomial expansion.
Expansion of Binomials: Understanding how to expand a binomial using the binomial theorem.
Binomial Coefficients: Understanding what binomial coefficients are and how to calculate them using factorial notation or Pascal's Triangle.
The General Binomial Theorem: Understanding the general formula for the binomial theorem, which enables you to expand any power of a binomial.
Applications of the Binomial Theorem: Understanding how the binomial theorem can be used to solve problems and carry out calculations in probability theory and other areas of mathematics.
Multinomial Theorem: Understanding the expanded version of the binomial theorem, the multinomial theorem, which enables you to expand expressions with more than two terms.
Generating Functions: Understanding what generating functions are and how they are related to the binomial theorem.
Advanced Topics: Advanced topics related to the binomial theorem, such as the Vandermonde identity, the combinatorial identity, and the Stirling numbers, among others.
Binomial coefficient: A mathematical expression that represents the number of ways to choose k items from a set of n items. This is given by the formula n choose k or C(n,k).
Binomial expansion: A formula that allows one to expand a binomial expression raised to a power using the binomial coefficients. It is written as (a + b)^n = ∑[k=0 to n] C(n,k) a^(n-k) b^k.
Binomial distribution: A probability distribution that denotes the number of successes in a fixed number of independent trials with two possible outcomes.
Multinomial coefficient: A mathematical expression that represents the number of ways to distribute n objects among k distinct classes. It is given by the formula (n choose n1, n2, ..., nk) = n! / (n1! × n2! × ... × nk!).
Multinomial theorem: A formula that allows one to expand a multinomial expression raised to a power using the multinomial coefficients. It is written as (a1 + a2 + ... + ak)^n = ∑[n1 + n2 + ... + nk = n] (n choose n1, n2, ..., nk) a1^(n1) a2^(n2) ... ak^(nk).
Negative binomial distribution: A probability distribution that denotes the number of failures before a fixed number of successes is achieved in a sequence of independent trials with two possible outcomes.
Trinomial coefficient: A mathematical expression that represents the number of ways to distribute n objects among three distinct classes. It is given by the formula (n choose p, q, r) = n! / (p! × q! × r!), where p + q + r = n.
Trinomial theorem: A formula that allows one to expand a trinomial expression raised to a power using the trinomial coefficients. It is written as (a + b + c)^n = ∑[p+q+r=n] (n choose p,q,r) a^p b^q c^r.
Pascal's triangle: A triangular array of numbers where each number is the sum of the two numbers directly above it. The binomial coefficients are arranged in this triangle.
Vandermonde's identity: A formula that expresses the sum of certain binomial coefficients as a single binomial coefficient. It is written as (m+n choose k) = ∑[r=0 to k] (m choose r) (n choose k-r).
"It is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc."
"The exponents b and c are nonnegative integers with b + c = n."
"The coefficient a in the term of axbyc is known as the binomial coefficient."
"The coefficient a in the term of axbyc is known as the binomial coefficient (n choose b) or (n choose c)."
"These coefficients for varying n and b can be arranged to form Pascal's triangle."
"These numbers also occur in combinatorics."
"(n choose b) gives the number of different combinations of b elements that can be chosen from an n-element set."
"(n choose b) is often pronounced as 'n choose b'."
"The coefficient a of each term is a specific positive integer depending on n and b."
"The exponents b and c are nonnegative integers."
"The exponents b and c are such that b + c = n."
"The polynomial (x + y)n can be expanded into a sum involving terms of the form axbyc."
"The binomial expansion describes the algebraic expansion of powers of a binomial in elementary algebra."
"The binomial coefficient determines the coefficient of each term in the binomial expansion."
"The expansion of (x + y)n involves terms of the form axbyc."
"The binomial coefficient (n choose b) corresponds to the number of different combinations of b elements that can be chosen from an n-element set."
"The exponents b and c represent the distribution of powers in the expanded terms."
"The binomial coefficient can have different values depending on n and b."
"The expansion of (x + y)n is useful in combinatorics and can be applied to solve problems involving combinations and permutations."