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.
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The binomial theorem states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.
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.
Pascal's Triangle can be used to find the coefficients for each term in the binomial expansion.
Each term in the binomial expansion has the form $\binom{n}{k} a^{n-k} b^k$, where k ranges from 0 to n.
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 $(x + y)^5$?