Binomial Theorem
Definition
The Binomial Theorem provides a formula to expand any binomial expression raised to a positive integer power. It expresses the expansion as a sum involving terms of the form $a^k b^{n-k}$ multiplied by binomial coefficients.
5 Must Know Facts For Your Next Test
- The general form of the Binomial Theorem is $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}$.
- Binomial coefficients are calculated using $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $!$ denotes factorial.
- Pascal's Triangle can be used to find the binomial coefficients directly without computation.
- Each term in the expansion of $(a + b)^n$ has exactly one term for each combination of powers of $a$ and $b$ that add up to $n$.
- The Binomial Theorem is useful in probability theory for expanding expressions like $(p+q)^n$, which represent probabilities in binomial distributions.
Review Questions
- What is the expanded form of $(x + y)^3$ using the Binomial Theorem?
- How do you compute the binomial coefficient $\binom{5}{2}$?
- Explain how Pascal's Triangle helps in finding binomial coefficients.
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Related terms
Factorial: The product of all positive integers up to a given number n, denoted by n!. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it. It represents binomial coefficients.
Binomial Coefficient: $\binom{n}{k}$ represents the number of ways to choose k elements from an n-element set. Calculated as $\frac{n!}{k!(n-k)!}$.
