(a+b)^10 Formula

5 min read Jun 16, 2024

Understanding the Binomial Theorem: (a + b)^10

The expression (a + b)^10 might seem daunting at first glance, but it can be expanded using the Binomial Theorem. This powerful theorem allows us to expand any binomial raised to a positive integer power.

The Binomial Theorem Formula

The general formula for the Binomial Theorem is:

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

where:

n is the power to which the binomial is raised
k is an integer ranging from 0 to n
(n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!), which represents the number of ways to choose k items from a set of n items.

Expanding (a + b)^10

To expand (a + b)^10 using the Binomial Theorem, we need to calculate the terms for each value of k from 0 to 10.

Let's break it down:

k = 0: (10 choose 0) * a^(10-0) * b^0 = 1 * a^10 * 1 = a^10
k = 1: (10 choose 1) * a^(10-1) * b^1 = 10 * a^9 * b = 10a^9b
k = 2: (10 choose 2) * a^(10-2) * b^2 = 45 * a^8 * b^2 = 45a^8b^2
k = 3: (10 choose 3) * a^(10-3) * b^3 = 120 * a^7 * b^3 = 120a^7b^3
k = 4: (10 choose 4) * a^(10-4) * b^4 = 210 * a^6 * b^4 = 210a^6b^4
k = 5: (10 choose 5) * a^(10-5) * b^5 = 252 * a^5 * b^5 = 252a^5b^5
k = 6: (10 choose 6) * a^(10-6) * b^6 = 210 * a^4 * b^6 = 210a^4b^6
k = 7: (10 choose 7) * a^(10-7) * b^7 = 120 * a^3 * b^7 = 120a^3b^7
k = 8: (10 choose 8) * a^(10-8) * b^8 = 45 * a^2 * b^8 = 45a^2b^8
k = 9: (10 choose 9) * a^(10-9) * b^9 = 10 * a^1 * b^9 = 10ab^9
k = 10: (10 choose 10) * a^(10-10) * b^10 = 1 * a^0 * b^10 = b^10

Therefore, the expanded form of (a + b)^10 is:

a^10 + 10a^9b + 45a^8b^2 + 120a^7b^3 + 210a^6b^4 + 252a^5b^5 + 210a^4b^6 + 120a^3b^7 + 45a^2b^8 + 10ab^9 + b^10

Pascal's Triangle

A handy tool for calculating binomial coefficients is Pascal's Triangle. Each row of Pascal's Triangle represents the coefficients of the expanded binomial for a given power. The numbers in the triangle are formed by adding the two numbers directly above them.

Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1 ... Row 10: 1 10 45 120 210 252 210 120 45 10 1

Notice that the coefficients in the expansion of (a + b)^10 match the numbers in the 10th row of Pascal's Triangle.

Conclusion

The Binomial Theorem provides a systematic way to expand binomials raised to any power. Understanding the theorem and its applications is crucial in various fields like algebra, calculus, and probability. While the expansion of (a + b)^10 might seem complex, using the formula and Pascal's Triangle can make the process much easier.