How Does Pascal’s Triangle Relate to the Binomial Theorem?

Pascal’s Triangle is a triangular array of numbers that reflects the coefficients in the expansion of a binomial expression. The relationship between Pascal’s Triangle and the Binomial Theorem is quite profound and elegantly illustrated.

The Binomial Theorem states that for any non-negative integer n, the expansion of the binomial expression (a + b)ⁿ can be expressed as:

(a + b)ⁿ = Σ (n choose k) * a^k * b^n-k

Here, n choose k, denoted as C(n, k) or n! / (k! * (n – k)!), represents the number of ways to choose k elements from a set of n elements. The coefficients in front of a^k and b^n-k are exactly the numbers found in Pascal’s Triangle.

For example, if we want to expand (a + b)³, we can use Pascal’s Triangle to find the coefficients:

• Row 0: 1
• Row 1: 1, 1
• Row 2: 1, 2, 1
• Row 3: 1, 3, 3, 1

This tells us that:

(a + b)³ = 1*a³*b⁰ + 3*a²*b¹ + 3*a¹*b² + 1*a⁰*b³ = a³ + 3a²b + 3ab² + b³

In summary, each row of Pascal’s Triangle corresponds to the coefficients of the binomial expansion for increasing values of n. Thus, Pascal’s Triangle provides a simple and systematic way to find the coefficients required for the Binomial Theorem, effectively connecting these two important mathematical concepts.