Pascal's Triangle

Definition

Pascal's Triangle is a triangular array of binomial coefficients. Each number is the sum of the two directly above it.

5 Must Know Facts For Your Next Test

1. The $n$-th row of Pascal's Triangle corresponds to the coefficients in the expansion of $(a + b)^n$.
2. The elements in Pascal's Triangle are symmetrical.
3. Each entry in Pascal’s Triangle can be calculated using the formula $\binom{n}{k}$, where $n$ is the row number and $k$ is the position within the row.
4. The sum of the elements in the $n$-th row is equal to $2^n$.
5. Pascal's Triangle can be used to find combinations, which is fundamental in probability and counting theory.

Review Questions

• How do you determine an entry in Pascal’s Triangle using binomial coefficients?
• What does the $10$th row of Pascal’s Triangle represent in terms of binomial expansion?
• Why are the rows of Pascal’s Triangle symmetrical?

Related terms

Binomial Coefficient: A coefficient that appears in a binomial expansion, represented as $\binom{n}{k}$ and calculated as $\frac{n!}{k!(n-k)!}$.

Combination: $C(n, k)$ represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.

$(a + b)^n$: $(a + b)^n$ represents a binomial expression raised to a power, where each term in its expansion corresponds to an entry in Pascal’s Triangle.