10.7
Binomial Expansion Using Pascal's Triangle
A binomial is an expression of the form a + b, where a and b are numbers or algebraic expressions.
Raising it to a power n produces a series of terms that follow a predictable pattern.
Each expansion has n + 1 terms, starting with an and ending with bn.
These patterns align with a visual tool known as Pascal’s Triangle.
Pascal’s Triangle is a triangular array where each row provides the coefficients for a binomial raised to a specific power.
For example, the fifth row gives the coefficients for a + b raised to the fifth power.
Each row begins and ends with one, and each inner number equals the sum of the two numbers diagonally above it. Pascal’s triangle provides the coefficients of the Binomial Theorem, calculated as n choose k.
This pattern also applies to probability. In coin tosses, H and T represent heads and tails. For three tosses, the sum of H and T raised to the third power represents all possible outcomes.
After expanding and comparing with Pascal’s triangle, each term corresponds to a possible outcome: three heads, two heads and one tail, one head and two tails, or three tails.
Expanding a binomial expression such as (a + b)n results in a predictable sequence of terms that can be systematically derived using Pascal’s Triangle. This triangular array of numbers plays a central role in understanding and computing the coefficients of binomial expansions.
Pascal’s Triangle is constructed such that each row corresponds to the coefficients of a binomial raised to a power. The topmost row, known as the zeroth row, corresponds to (a + b)0, and each successive row gives the coefficients for increasing powers of n. For example, the sixth row of Pascal’s Triangle, 1, 5, 10, 10, 5, 1, represents the coefficients in the expansion of (a + b)5. Each row starts and ends with the number 1. Every inner entry is computed by summing the two entries diagonally above it in the preceding row, illustrating the recursive structure of the triangle.
In the expansion of (a + b)n, the exponents of a decrease from n to 0, while the exponents of b increase from 0 to n. Consequently, each term in the expansion takes the form:
Where “n choose r” denotes the binomial coefficient, found at the rth position of the nth row in Pascal’s Triangle.
A binomial is an expression of the form a + b, where a and b are numbers or algebraic expressions.
Raising it to a power n produces a series of terms that follow a predictable pattern.
Each expansion has n + 1 terms, starting with an and ending with bn.
These patterns align with a visual tool known as Pascal’s Triangle.
Pascal’s Triangle is a triangular array where each row provides the coefficients for a binomial raised to a specific power.
For example, the fifth row gives the coefficients for a + b raised to the fifth power.
Each row begins and ends with one, and each inner number equals the sum of the two numbers diagonally above it. Pascal’s triangle provides the coefficients of the Binomial Theorem, calculated as n choose k.
This pattern also applies to probability. In coin tosses, H and T represent heads and tails. For three tosses, the sum of H and T raised to the third power represents all possible outcomes.
After expanding and comparing with Pascal’s triangle, each term corresponds to a possible outcome: three heads, two heads and one tail, one head and two tails, or three tails.
From Chapter 10:
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