Binomial Theorem Flashcard

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What is the Binomial Theorem?

Back

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 = 0 to n.

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What does 'n choose k' mean in the context of the Binomial Theorem?

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'n choose k', denoted as C(n, k) or nCk, represents the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated as: C(n, k) = n! / (k!(n-k)!).

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How do you find the coefficient of a specific term in a binomial expansion?

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To find the coefficient of a term a^k * b^(n-k) in the expansion of (a + b)^n, use the formula: Coefficient = C(n, k) * a^k * b^(n-k).

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What is Pascal's Triangle?

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Pascal's Triangle is a triangular array of the binomial coefficients. Each number is the sum of the two directly above it, and it provides a quick way to find coefficients for binomial expansions.

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How many rows are in Pascal's Triangle?

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Pascal's Triangle has an infinite number of rows.

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What is the 6th term of the expansion of (m - 3)^5?

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The 6th term is -243.

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Find the 4th term in the expansion of (4x - 3)^5.

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The 4th term is -4320x^2.

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What is the third term for the expansion (2x - 5)^5?

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The third term is 2000x^3.

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Calculate the coefficient of a^4b^6 in the expansion of (a + b)^{10}.

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The coefficient is 210.

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What is the general form of a term in the expansion of (a + b)^n?

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The general term T(k+1) in the expansion of (a + b)^n is given by T(k+1) = C(n, k) * a^(n-k) * b^k.