Binomial coefficient

Definition

A binomial coefficient, denoted as $\binom{n}{k}$, represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order. It is an essential component in the expansion of binomial expressions.

5 Must Know Facts For Your Next Test

1. The formula for calculating a binomial coefficient is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
2. Binomial coefficients are used in the Binomial Theorem to expand expressions of the form $(a+b)^n$.
3. $\binom{n}{k}$ is equal to $\binom{n}{n-k}$ due to symmetry.
4. Binomial coefficients can be found in Pascal's Triangle; each entry is the sum of the two directly above it.
5. They are also used in combinatorial problems and probability calculations.

Review Questions

• What is the binomial coefficient $\binom{5}{2}$ and how do you calculate it?
• Explain how binomial coefficients are related to Pascal's Triangle.
• How does the Binomial Theorem use binomial coefficients?

Related terms

Binomial Theorem:

A theorem that provides a formula for expanding powers of binomials: $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.

Factorial: $n!$ (read as 'n factorial') is the product of all positive integers up to n; used in calculating permutations and combinations.

Pascal's Triangle:

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