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from class: College Algebra 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.
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Predict what's on your test 5 Must Know Facts For Your Next Test The binomial coefficient $\binom{n}{k}$ can be calculated using the formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. Binomial coefficients are symmetric, meaning $\binom{n}{k} = \binom{n}{n-k}$. They appear as entries in Pascal's Triangle, where each entry is the sum of the two directly above it. In the Binomial Theorem, they are used to expand expressions of the form $(a + b)^n$ into a sum involving terms like $\binom{n}{k}a^{n-k}b^k$. Binomial coefficients have applications in combinatorics, probability theory, and algebra. Review Questions What is the formula for calculating a binomial coefficient? How do you interpret $\binom{6}{2}$ in terms of choosing elements? Explain how binomial coefficients are related to Pascal's Triangle. "Binomial coefficient" also found in:
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