PASCAL'S TRIANGLE AND BINOMIAL THEOREM

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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 starts with a '1' at the top, and the rows correspond to the coefficients of the binomial expansion.

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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 a binomial. It states that: $$(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k$$ where $${n \choose k}$$ is a binomial coefficient.

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What is the formula for the binomial coefficient?

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The binomial coefficient is given by: $${n \choose k} = \frac{n!}{k!(n-k)!}$$ where $n!$ (n factorial) is the product of all positive integers up to n.

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Expand: $$(a + b)^2$$

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$$(a + b)^2 = a^2 + 2ab + b^2$$

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Expand: $$(a - b)^3$$

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$$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$

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What is the 4th term in the expansion of $$(x + y)^5$$?

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The 4th term is given by $${5 \choose 3} x^{5-3} y^3 = 10x^2y^3$$.

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

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The number 1 appears at the edges of Pascal's Triangle, representing the coefficients of the first and last terms in the binomial expansion.

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How do you find the nth term in the expansion of $$(a + b)^n$$?

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The nth term can be found using the formula: $${n \choose k} a^{n-k} b^k$$ where k is the term number starting from 0.

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

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

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What is the 5th term in the expansion of $$(m - 3)^5$$?

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The 5th term is $${5 \choose 4} m^{5-4} (-3)^4 = 5m(81) = 405m$$.