Pascal's Triangle Exploration

Pascal's Triangle Exploration

Read each question carefully and provide your answers in the space provided. Show all your work where applicable.

Refer to the diagram below for questions 1-3. It shows the first few rows of Pascal's Triangle.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1. The sum of the numbers in row 4 (starting count from row 0) of Pascal's Triangle is  .

2. The numbers in row 3 of Pascal's Triangle are the coefficients for the binomial expansion of (a + b)³ which is  .

3. The third number in row 6 (starting count from row 0) of Pascal's Triangle is  .

4. Using Pascal's Triangle, expand the binomial expression (x + y)⁴.

5. Determine the coefficient of the x³y² term in the expansion of (x + y)⁵.

6. Which row of Pascal's Triangle corresponds to the coefficients of (a + b)⁶?

7. How many different combinations are there to choose 3 items from a set of 5 distinct items? (Hint: Think about Pascal's Triangle)

8. The numbers along the diagonals of Pascal's Triangle represent the natural numbers, triangular numbers, and tetrahedral numbers.

9. Every number in Pascal's Triangle is the sum of the two numbers directly above it.

10. Explain the relationship between Pascal's Triangle and combinations (nCr). Provide an example to illustrate your explanation.