Binomial Theorem

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What is the Binomial Theorem?

Back

The Binomial Theorem provides a formula for the expansion of powers of a binomial (a + b)^n, expressed as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.

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What is a binomial coefficient?

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A binomial coefficient, denoted as C(n, k) or '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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What is Pascal's Triangle?

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Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. It is used to find binomial coefficients.

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How do you find the 7th row of Pascal's Triangle?

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The 7th row of Pascal's Triangle corresponds to the coefficients of the expansion of (a + b)^7, which are 1, 7, 21, 35, 35, 21, 7, 1.

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What is the formula for the nth term of a binomial expansion?

Back

The nth term of the expansion of (a + b)^n is given by T(k+1) = C(n, k) * a^(n-k) * b^k, where k is the term number starting from 0.

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

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The 2nd term is 256y.

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What is the 5th term of the expansion of (n + 4)^4?

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The 5th term is 256.

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Expand (x - 7)^2.

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

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What is the 4th term of the expansion of (a + 3)^4?

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The 4th term is 108a.

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What is the general term in the expansion of (x + y)^n?

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The general term is T(k+1) = C(n, k) * x^(n-k) * y^k.