Binomial Theorem

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What is the Binomial Theorem?

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The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k for k = 0 to n.

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What does (n choose k) represent in 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 n! / (k!(n-k)!).

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What is Pascal's Triangle?

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

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What is the coefficient of x^k in the expansion of (a + b)^n?

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The coefficient of x^k in the expansion of (a + b)^n is given by C(n, k) * a^(n-k) * b^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 a binomial expansion, identify the values of n, a, b, and k, then use the formula C(n, k) * a^(n-k) * b^k.

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What is the significance of the negative sign in binomial expansions?

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The negative sign in binomial expansions indicates that the term is subtracted rather than added. For example, in (a - b)^n, the terms alternate in sign.

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Calculate the coefficient of x^9 in (3x - 1)^12.

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The coefficient of x^9 in (3x - 1)^12 is -4330260.

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What is row 5 of Pascal's Triangle?

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Row 5 of Pascal's Triangle is 1, 5, 10, 10, 5, 1.

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Find the coefficient of x^3 in (x - 3)^10.

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The coefficient of x^3 in (x - 3)^10 is -262440.

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Find the coefficient of x^2 in (2x - 3)^9.

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The coefficient of x^2 in (2x - 3)^9 is -314928.