key term - Binomial coefficient

Definition

A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.

5 Must Know Facts For Your Next Test

1. The binomial coefficient $\binom{n}{k}$ can be calculated using the formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
2. Binomial coefficients are symmetric, meaning $\binom{n}{k} = \binom{n}{n-k}$.
3. They appear as entries in Pascal's Triangle, where each entry is the sum of the two directly above it.
4. In the Binomial Theorem, they are used to expand expressions of the form $(a + b)^n$ into a sum involving terms like $\binom{n}{k}a^{n-k}b^k$.
5. Binomial coefficients have applications in combinatorics, probability theory, and algebra.

Review Questions

Related terms

Factorial:

The product of all positive integers up to a given number n, denoted by n!.

Pascal's Triangle:

A triangular array where each entry is the sum of the two directly above it and represents binomial coefficients.

Binomial Theorem:

$(a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$, which provides a way to expand powers of binomials.