What is the Binomial Theorem? Provide an example.

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∻Solution by Steps∻ ‖ step 1 ⋮ The Binomial Theorem provides a formula for expanding expressions that are raised to a power, specifically binomials of the form $ (a + b)^n $. The theorem states that: $$ (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k $$ where $ {n \choose k} $ is the binomial coefficient, which can be calculated as $ \frac{n!}{k!(n-k)!} $. ‖ ‖ step 2 ⋮ As an example, let's expand $ (x + y)^3 $ using the Binomial Theorem. ‖ ‖ step 3 ⋮ Apply the formula: $$ (x + y)^3 = \sum_{k=0}^{3} {3 \choose k} x^{3-k} y^k $$ ‖ ‖ step 4 ⋮ Calculate the terms individually: $$ {3 \choose 0}x^3y^0 + {3 \choose 1}x^2y^1 + {3 \choose 2}x^1y^2 + {3 \choose 3}x^0y^3 $$ ‖ ‖ step 5 ⋮ Simplify the binomial coefficients and the terms: $$ 1 \cdot x^3 \cdot 1 + 3 \cdot x^2 \cdot y + 3 \cdot x \cdot y^2 + 1 \cdot 1 \cdot y^3 $$ ‖ ‖ step 6 ⋮ Combine the terms to get the final expanded form: $$ x^3 + 3x^2y + 3xy^2 + y^3 $$ ‖ ∻Answer∻ ⚹ The expanded form of $ (x + y)^3 $ using the Binomial Theorem is $ x^3 + 3x^2y + 3xy^2 + y^3 $ ⚹ ∻Key Concept∻ ⚹The Binomial Theorem is a formula for expanding binomials raised to a power.⚹ ∻Explanation∻ ⚹The theorem expresses the expansion as the sum of terms involving binomial coefficients, which are determined by the formula $ {n \choose k} = \frac{n!}{k!(n-k)!} $. Each term in the expansion is a product of the binomial coefficients, powers of the first term, and powers of the second term.⚹◊How is the Binomial Theorem used in calculating the probabilities of specific⍭ Generate me a similar question◊

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