Question

Write the eighth row of Pascal's Triangle as combinations and as numbers.

Short answer

The eighth row of Pascal's Triangle in terms of combinations is: \(C(8,0)\), \(C(8,1)\), \(C(8,2)\), \(C(8,3)\), \(C(8,4)\), \(C(8,5)\), \(C(8,6)\), \(C(8,7)\), \(C(8,8)\). In terms of numbers, it is: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Step-by-step solution

Understand Pascal Triangle Pattern

A Pascal Triangle is a geometric figure formed by arrangements of numbers. It starts with '1' at the top. Each number below is the sum of the two digits directly above it (right and left). In the term of combinations, each element is an answer of how many ways we can choose items out of a larger set.

Calculate the 8th Row as Combinations

The 8th row of the Pascal Triangle (actually the 9th row, since Pascal's Triangle starts with row 0), corresponds to the coefficients of the terms in the binomial expansion of (a+b) to the power of 8, written as \(C(n,r) = C(8,r)\). So the 8th row in terms of combinations is: \(C(8,0)\), \(C(8,1)\), \(C(8,2)\), \(C(8,3)\), \(C(8,4)\), \(C(8,5)\), \(C(8,6)\), \(C(8,7)\), \(C(8,8)\)

Calculate 8th Row as Numbers

Using the combination formula, the 8th row of Pascal's Triangle is: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Key concepts

Combinations

When studying mathematics, particularly probability and algebra, the concept of combinations plays a pivotal role. Simply put, a combination is a selection of items from a larger set where the order does not matter. In Pascal's Triangle, combinations can be used to represent the coefficients in the expansion of binomials.

Consider picking a team of 3 people from a group of 10. The formula for combinations, denoted as \( C(n, r) \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose, allows us to calculate all possible selections. In the context of Pascal's Triangle, the eighth row represents all the combinations from 0 to 8: \( C(8,0), C(8,1), ..., C(8,8) \). The value of each combination speaks to the number of ways to choose \( r \) elements from a set of \( n \) without regard to the order. This is crucial for understanding the binomial theorem and, by extension, the nature of Pascal's Triangle itself.

Binomial Expansion

At the heart of algebra lies the binomial expansion, which provides a way to express the power of a binomial, like \( (a+b)^n \), as a sum of simpler terms. Pascal's Triangle has a remarkable connection to this process, as each row of the triangle signifies the coefficients of the terms in the expansion.

Let's consider the binomial \( (a+b)^8 \). To expand this expression, we are essentially looking for the coefficients that accompany each term when multiplied out. These coefficients can be cumbersome to calculate with sheer multiplication as the power increases, which is where Pascal's Triangle shines. By referencing the eighth row (1, 8, 28, 56, 70, 56, 28, 8, 1), we can find each coefficient relevant to the expanded form: \( a^8 + 8a^7b + 28a^6b^2 + ... + b^8 \).

These coefficients match the sequence of numbers in the eighth row, thus demonstrating how the triangle simplifies the expansion process. Understanding this helps not just in theoretical algebra but also in practical applications such as calculating probabilities where binomial distributions are concerned.

Geometric Figure

Pascal's Triangle is not just a mathematical concept but also a geometric figure that can be visualized as a pyramid of numbers. Starting with a solitary '1' at the top, each subsequent row builds upon the previous one with numbers that are the sums of the two numbers situated diagonally above.

This geometric representation is incredibly intuitive: looking at the triangle, one can easily spot patterns and properties, such as its symmetrical nature or how each edge is formed exclusively of '1's. The geometric aspect makes Pascal's Triangle a particularly engaging tool for learning combinatorics and binomial expansions.

Moreover, by regarding the triangle as a geometric figure, educators can adopt hands-on teaching techniques, like constructing the triangle with coins or counters to illuminate the concept further. It also knits a relationship between visual patterns and algebraic concepts, allowing learners to better remember and apply mathematical principles in various contexts.