Pascal's Triangle
Use Pascal Triangle in Grade 10 algebra and combinatorics. Build rows by summing adjacent entries, extract binomial coefficients, and apply the triangle to expand (a+b)ⁿ quickly.
Key Concepts
Pascal's Triangle is a triangular array where each number is the sum of the two directly above it. The numbers in the nth row are the coefficients for the terms in the expansion of the binomial $(a+b)^n$. The top row is considered the zero row.
Expand $(x+y)^3$: Use row 3 (1, 3, 3, 1) to get $1x^3 + 3x^2y + 3xy^2 + 1y^3$. Expand $(a+b)^4$: Use row 4 (1, 4, 6, 4, 1) to get $a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$.
Think of this triangle as a visual shortcut for expanding binomials. Instead of tediously multiplying $(a+b)$ by itself, you just find the correct row for your power and use those numbers as coefficients. It’s a handy map that lays out all the numbers you need to quickly solve the expansion problem.
Common Questions
How do you construct Pascal Triangle?
Each row begins and ends with 1. Every interior number equals the sum of the two numbers directly above it. Row n contains the binomial coefficients for expanding (a + b)ⁿ.
How does Pascal Triangle relate to the binomial theorem?
The nth row (starting from row 0) gives the coefficients for (a + b)ⁿ. For (a + b)³, use row 3: 1, 3, 3, 1, giving a³ + 3a²b + 3ab² + b³.
How are Pascal Triangle entries related to combinations?
The entry in row n, position k equals C(n,k) = n!/(k!(n-k)!). Row 4 position 2 = C(4,2) = 6. Pascal Triangle is a visual representation of all combination values.