Binomial Theorem
Expansion of (x+y)ⁿ
About Binomial Theorem
The Binomial Theorem represents expansion of (x+y)ⁿ. This algebra formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Algebra and Combinatorics. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Binomial Theorem include Probability theory, Statistics, Combinatorics, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^kFormula Information
Difficulty Level
Prerequisites
Discovered
1665
Discoverer
Isaac Newton
Real-World Applications
Examples
Mathematical Fields
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Important Notes
The coefficients are given by binomial coefficients C(n,k) = n!/(k!(n-k)!) = (n choose k). These coefficients form Pascal's triangle. For negative or fractional exponents, use the generalized binomial theorem: (1+x)ᵃ = Σ C(a,k) xᵏ for |x| < 1. The sum of all coefficients is 2ⁿ.
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Formula Variations
Frequently Asked Questions
What is a binomial coefficient?
A binomial coefficient, denoted as C(n,k) or (n choose k), represents the number of ways to choose k items from n items without regard to order. It's calculated as n!/(k!(n-k)!). For example, C(5,2) = 10 means there are 10 ways to choose 2 items from 5.
How do I find a specific term in a binomial expansion?
The (k+1)th term in the expansion of (x+y)ⁿ is given by T_{k+1} = C(n,k) x^(n-k) y^k. For example, to find the 3rd term of (x+y)⁵, use k=2: T₃ = C(5,2) x³ y² = 10x³y².
What is Pascal's triangle and how does it relate to the binomial theorem?
Pascal's triangle is a triangular array where each number is the sum of the two numbers directly above it. The nth row of Pascal's triangle contains the binomial coefficients for (x+y)ⁿ. For example, row 4 is 1, 4, 6, 4, 1, which are the coefficients of (x+y)⁴.
Can the binomial theorem be used for negative or fractional exponents?
Yes, the generalized binomial theorem works for any real exponent a: (1+x)ᵃ = Σ C(a,k) xᵏ, where the sum is from k=0 to infinity. However, this infinite series only converges when |x| < 1. The binomial coefficients are calculated using the gamma function for non-integer values.
What is the sum of all coefficients in a binomial expansion?
The sum of all coefficients in (x+y)ⁿ is 2ⁿ. This can be seen by setting x = 1 and y = 1 in the expansion, which gives (1+1)ⁿ = 2ⁿ. For example, the coefficients of (x+y)³ are 1, 3, 3, 1, and their sum is 8 = 2³.
How is the binomial theorem used in probability?
The binomial theorem is fundamental in probability theory, particularly for the binomial distribution. If an event has probability p of success and (1-p) of failure, the probability of exactly k successes in n trials is given by C(n,k) pᵏ (1-p)ⁿ⁻ᵏ, which follows directly from the binomial theorem.
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Quick Details
- Category
- Algebra
- Difficulty
- Intermediate
- Discovered
- 1665
- Discoverer
- Isaac Newton
- Formula ID
- binomial