Pascal's Triangle Exploration

Pascal's Triangle Exploration

Carefully read each question 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, Persia, China, and Italy.

The rows of Pascal's triangle are conventionally numbered starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.

1. The 0th row of Pascal's Triangle contains a single number, which is  .

2. Each number in Pascal's Triangle is the sum of the two numbers directly   it.

3. The numbers along the outside edges of Pascal's Triangle are all  .

4. Complete the first 5 rows (row 0 to row 4) of Pascal's Triangle below:

5. What is the sum of the numbers in Row 4 of Pascal's Triangle?

6. The numbers in the second diagonal (from either side) of Pascal's Triangle represent:

7. How can Pascal's Triangle be used to find the coefficients in the expansion of a binomial expression like (x + y)^n?

8. Use Pascal's Triangle to expand the expression (2a + b)^3.

9. The sum of the numbers in any row of Pascal's Triangle is a power of 2.

10. Each row of Pascal's Triangle is symmetrical.