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 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 is used to find coefficients in the expansion of binomials.

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How do you find the coefficients in the expansion of (a + b)^n using Pascal's Triangle?

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The coefficients of the expansion of (a + b)^n correspond to the nth row of Pascal's Triangle.

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What is the formula for the binomial coefficient, often denoted as 'n choose k'?

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The binomial coefficient is given by the formula C(n, k) = n! / (k!(n-k)!), where n! is the factorial of n.

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What is row 0 of Pascal's Triangle?

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Row 0 of Pascal's Triangle is simply 1.

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What is the significance of row 1 in Pascal's Triangle?

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Row 1 of Pascal's Triangle is 1, 1, which corresponds to the coefficients of (a + b)^1.

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Expand the expression (x + 2)^3 using the Binomial Theorem.

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The expansion is x^3 + 6x^2 + 12x + 8.

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What is the value of (3 + 4)^2 using the Binomial Theorem?

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Using the Binomial Theorem, (3 + 4)^2 = 3^2 + 2(3)(4) + 4^2 = 9 + 24 + 16 = 49.

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What is the relationship between the rows of Pascal's Triangle and the powers of binomials?

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Each row of Pascal's Triangle corresponds to the coefficients of the expanded form of (a + b)^n, where n is the row number.

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How do you calculate the 5th row of Pascal's Triangle?

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The 5th row of Pascal's Triangle is calculated as 1, 5, 10, 10, 5, 1, which corresponds to the coefficients of (a + b)^5.