The row numbers in Pascal's Triangle represent the coefficients of the binomial expansion. Specifically, the $n$-th row of Pascal's Triangle contains the coefficients of the terms in the expansion of $(a + b)^n$. These coefficients are also known as binomial coefficients.

For example:

• The 0th row represents $(a + b)^0 = 1$, which is just [1].
• The 1st row represents $(a + b)^1 = a + b$, which is [1, 1].
• The 2nd row represents $(a + b)^2 = a^2 + 2ab + b^2$, which is [1, 2, 1].
• The 3rd row represents $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$, which is [1, 3, 3, 1].

The entries in each row can also be computed using combinations, denoted as $\binom{n}{k}$, where $n$ is the row number and $k$ ranges from 0 to $n$. These binomial coefficients are calculated using the formula:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How do you calculate a specific entry in Pascal's Triangle?
2. What is the connection between Pascal's Triangle and Fibonacci numbers?
3. Can you explain how Pascal's Triangle relates to the binomial theorem?
4. How is Pascal's Triangle used in probability and statistics?
5. What are some patterns that can be found in Pascal's Triangle?
6. How does Pascal's Triangle relate to polynomial expansion?
7. What is the historical significance of Pascal's Triangle?
8. How can Pascal's Triangle be used in combinatorics problems?

Tip: Pascal's Triangle can also be used to find the coefficients in the expansion of negative and fractional powers, providing insights into more advanced mathematical concepts.