Binomial Theorem

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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 binomials. It states that (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?

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The coefficients are given by the binomial coefficients, which can be found in Pascal's Triangle or calculated using the formula n choose k.

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How do you find the coefficients of the 5th row in Pascal's Triangle?

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The coefficients of the 5th row are found by calculating the binomial coefficients for n=5: 1, 5, 10, 10, 5, 1.

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Expand (x - 7)^2 using the Binomial Theorem.

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The expansion is x^2 - 14x + 49.

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What is the 2nd term in the expansion of (y + 4)^4?

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The 2nd term is 16y^3.

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What is row 7 of Pascal's Triangle?

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Row 7 is 1, 7, 21, 35, 35, 21, 7, 1.

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

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The expansion is x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.

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What is the formula for the binomial coefficient?

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The binomial coefficient is given by n choose k = n! / (k!(n-k)!).

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What does the term 'n choose k' represent?

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'n choose k' represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.

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How can you use Pascal's Triangle to find binomial coefficients?

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Each row of Pascal's Triangle corresponds to the coefficients of the binomial expansion for increasing powers of (a + b).