Binomial expansion

Definition

Binomial expansion is the process of expanding an expression that is raised to a power, specifically in the form $(a + b)^n$. It utilizes the binomial theorem to express the expanded form as a sum involving binomial coefficients.

5 Must Know Facts For Your Next Test

1. The binomial theorem states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.
2. The binomial coefficient $\binom{n}{k}$, also written as $C(n, k)$ or 'n choose k', represents the number of ways to choose k elements from a set of n elements without regard to order.
3. Pascal's Triangle can be used to find the coefficients for each term in the binomial expansion.
4. Each term in the binomial expansion has the form $\binom{n}{k} a^{n-k} b^k$, where k ranges from 0 to n.
5. In probability and statistics, binomial expansions are often used to figure out probabilities for multiple trials.

Review Questions

• State and explain the binomial theorem formula.
• What is the value of $\binom{7}{3}$ and how is it used in binomial expansion?
• How can Pascal's Triangle be utilized in determining coefficients for $(x + y)^5$?

Related terms

Binomial Coefficient: A numerical factor that multiplies each term in a binomial expansion, denoted as $\binom{n}{k}$.

Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it, used to find coefficients in binomial expansions.

Combination: A selection of items without considering the order, represented by $C(n, k)$ or $\binom{n}{k}$.