Binomial Expansion

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What is the Binomial Theorem?

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The Binomial Theorem provides a formula for the expansion of powers of a binomial, expressed as (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n.

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What are the coefficients in the expansion of (x + y)^n derived from?

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The coefficients in the expansion of (x + y)^n are derived from Pascal's Triangle.

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How do you find the coefficients for the expansion of (x + y)^3?

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The coefficients for the expansion of (x + y)^3 are found in the 3rd row of Pascal's Triangle: 1, 3, 3, 1.

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What is the expansion of (x + y)^3?

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The expansion of (x + y)^3 is x^3 + 3x^2y + 3xy^2 + y^3.

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What is the expansion of (1 - x)^5?

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The expansion of (1 - x)^5 is 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5.

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What is the expansion of (1 - 2y)^3?

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The expansion of (1 - 2y)^3 is 1 - 6y + 12y^2 - 8y^3.

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What is the general form of the binomial expansion for (a + b)^n?

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The general form is (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k = 0 to n.

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What does (n choose k) represent in the binomial expansion?

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(n choose k) represents the binomial coefficient, calculated as n! / (k!(n-k)!), which gives the number of ways to choose k elements from a set of n elements.

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How can you derive the expansion of (x + y)^n using the Binomial Theorem?

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You can derive the expansion by applying the Binomial Theorem formula, substituting a with x and b with y, and calculating the coefficients using (n choose k).

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What is the significance of Pascal's Triangle in binomial expansion?

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Pascal's Triangle provides a quick way to find the coefficients for the binomial expansion without calculating factorials.