Pascal's Triangle & Binomial Expansion

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What is Pascal's Triangle?

Back

Pascal's Triangle is a triangular array of the binomial coefficients. Each number is the sum of the two directly above it, starting with a 1 at the top.

Tags

CCSS.HSA.APR.C.5

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What is the Binomial Theorem?

Back

The Binomial Theorem describes the algebraic expansion of powers of a binomial. It states that: (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k = 0 to n.

Tags

CCSS.HSA.APR.C.5

Front

What are the coefficients in the expansion of (a + b)^n?

Back

The coefficients are given by the binomial coefficients, which can be found in Pascal's Triangle.

Tags

CCSS.HSA.APR.C.5

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

Back

x^3 + 6x^2 + 12x + 8.

Tags

CCSS.HSA.APR.C.5

Front

What is the formula for the nth row of Pascal's Triangle?

Back

The nth row corresponds to the coefficients of (a + b)^n, represented as (n choose k) for k = 0 to n.

Tags

CCSS.HSA.APR.C.5

Front

What is the 4th row of Pascal's Triangle?

Back

1; 4; 6; 4; 1.

Tags

CCSS.HSA.APR.C.5

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How do you find the 3rd term in the expansion of (x + 4)^5?

Back

Use the formula: (5 choose 2) * x^(5-2) * 4^2 = 160x^3.

Tags

CCSS.HSA.APR.C.5

Front

What is the expansion of (a + b)^2?

Back

a^2 + 2ab + b^2.

Tags

CCSS.HSA.APR.C.5

Front

What is the significance of the coefficients in binomial expansion?

Back

The coefficients represent the number of ways to choose elements from a set, which is fundamental in combinatorics.

Tags

CCSS.HSA.APR.C.5

Front

How do you calculate (n choose k)?

Back

(n choose k) = n! / (k!(n-k)!), where n! is the factorial of n.