Pascal's Triangle and Binomial Expansion

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What is Pascal's Triangle?

Back

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 coefficients in binomial expansions.

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CCSS.HSA.APR.C.5

Front

What is the formula for binomial expansion?

Back

The binomial expansion of (a + b)^n is given by: \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k, where \binom{n}{k} is a binomial coefficient.

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CCSS.HSA.APR.C.5

Front

What is the binomial coefficient \binom{n}{k}?

Back

The binomial coefficient \binom{n}{k} represents the number of ways to choose k elements from a set of n elements, calculated as \frac{n!}{k!(n-k)!}.

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CCSS.HSA.APR.C.5

Front

How do you find the coefficients for (x + y)^4 using Pascal's Triangle?

Back

The coefficients for (x + y)^4 are found in the 4th row of Pascal's Triangle: 1, 4, 6, 4, 1.

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CCSS.HSA.APR.C.5

Front

Expand (x + 5)^2.

Back

(x + 5)^2 = x^2 + 10x + 25.

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CCSS.HSA.APR.C.5

Front

What is the 3rd row of Pascal's Triangle?

Back

The 3rd row of Pascal's Triangle is: 1, 3, 3, 1.

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CCSS.HSA.APR.C.5

Front

Expand (a + b)^3.

Back

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

Tags

CCSS.HSA.APR.C.5

Front

What is the significance of the diagonal sums in Pascal's Triangle?

Back

The sums of the diagonals in Pascal's Triangle correspond to the Fibonacci sequence.

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CCSS.HSA.APR.C.5

Front

Expand (x + 5)^5.

Back

(x + 5)^5 = x^5 + 25x^4 + 250x^3 + 1250x^2 + 3125x + 3125.

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CCSS.HSA.APR.C.5

Front

What pattern do you observe in the rows of Pascal's Triangle?

Back

Each row corresponds to the coefficients of the binomial expansion of (a + b)^n, where n is the row number starting from 0.

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CCSS.HSA.APR.C.5