Pascal's Triangle Exploration

Pascal's Triangle Exploration

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?

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.

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.