Binomial Theorem Flashcard

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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 = 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 specific term in the expansion of (a + b)^n, use the formula: Coefficient = C(n, k) * a^(n-k) * b^k, where k is the exponent of b in the term.

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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 in the expansion of (3x + 2)^8?

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The 6th term is 48384x^3.

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How do you find the coefficient of the x^5 term in (5x + 1)^{11}?

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The coefficient of the x^5 term is found using the formula: C(11, 5) * (5x)^5 * (1)^(11-5) = 1443750.

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What is the fourth term in the expansion of (\frac{x}{2} + y)^7?

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The fourth term is \frac{35x^4y^3}{16}.

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What are the coefficients of the 5th row in Pascal's Triangle?

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The coefficients of the 5th row are 1; 5; 10; 10; 5; 1.

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What is the general form of a term in the binomial 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.