Pascal's Triangle Exploration

A Grade 12 math worksheet exploring the properties and applications of Pascal's Triangle, including binomial expansion and combinatorial coefficients.

Grade 12 Math AlgebraPascal's Triangle

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Includes

TextFill in the BlanksMultiple Choice2 Short AnswerTrue / FalseMatching

Standards

CCSS.MATH.CONTENT.HSA.APR.C.5CCSS.MATH.CONTENT.HSS.CP.B.9

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Pascal's Triangle Exploration

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Read each question carefully and provide your answers in the space provided. Show all your work for full credit.

Pascal's Triangle is a triangular array of the binomial coefficients. It is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, China, Iran, Germany, and Italy. Each number is the sum of the two numbers directly above it.

Below are the first few rows of Pascal's Triangle:

Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1

1. Write out the next two rows (Row 5 and Row 6) of Pascal's Triangle below: Row 5: 1 5 10 10 5 1 Row 6: 1 6 15 20 15 6 1

2. What are the coefficients for the expansion of (a + b)⁵?

1, 4, 6, 4, 1

1, 5, 10, 10, 5, 1

1, 2, 1

1, 6, 15, 20, 15, 6, 1

3. Expand the binomial (x + y)⁴ using Pascal's Triangle.

4. The sum of the numbers in Row n of Pascal's Triangle is 2ⁿ.

True

False

5. Explain the relationship between the numbers in Pascal's Triangle and combinations (nCr). Provide an example.

Match each mathematical concept with its connection to Pascal's Triangle.

1. Binomial Theorem

a. Used to find the number of ways to choose k items from n items.

2. Combinations

b. Provides the coefficients for expanding (a+b)ⁿ.

3. Fibonacci Sequence

c. Found by summing the numbers along diagonals of the triangle.