Pascal's Triangle and Binomial Theorem

Front

What is Pascal's Triangle?

Back

Pascal's Triangle is a triangular array of the binomial coefficients, where each number is the sum of the two directly above it. It starts with a '1' at the top, and each subsequent row corresponds to the coefficients of the binomial expansion.

Front

How do you expand a binomial expression using the Binomial Theorem?

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The Binomial Theorem states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. The coefficients (n choose k) can be found in Pascal's Triangle.

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What is the formula for finding the number of terms in the expansion of (a + b)^n?

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The number of terms in the expansion of (a + b)^n is given by n + 1.

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What is the 14th term in the expansion of (a + b)^n?

Back

The 14th term in the expansion of (a + b)^n can be found using the formula T(k+1) = (n choose k) * a^(n-k) * b^k, where k = 13 for the 14th term.

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How do you find a specific element in Pascal's Triangle?

Back

To find a specific element in Pascal's Triangle, use the formula C(n, k) = n! / (k!(n-k)!), where n is the row number and k is the position in that row.

Front

What is the significance of the coefficients in the expansion of a binomial?

Back

The coefficients in the expansion of a binomial represent the number of ways to choose k elements from n elements, which corresponds to the binomial coefficients found in Pascal's Triangle.

Front

What is the relationship between binomial expansions and Pascal's Triangle?

Back

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

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How do you calculate (4x + 3y)^4 using the Binomial Theorem?

Back

Using the Binomial Theorem, (4x + 3y)^4 = Σ (4 choose k) * (4x)^(4-k) * (3y)^k for k = 0 to 4, resulting in 256x^4 + 768x^3y + 864x^2y^2 + 432xy^3 + 81y^4.

Front

What is the 8th row of Pascal's Triangle?

Back

The 8th row of Pascal's Triangle is 1, 8, 28, 56, 70, 56, 28, 8, 1.

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How do you find the 6th element in the 8th row of Pascal's Triangle?

Back

The 6th element in the 8th row of Pascal's Triangle is found using C(8, 5) = 28.