(a-b)^5 Pascal's Triangle

4 min read Jul 03, 2024

(a-b)^5 and Pascal's Triangle

Binomial Expansion and Pascal's Triangle

In algebra, the binomial theorem is a fundamental concept that describes the expansion of powers of a binomial, which is an expression consisting of two terms. One of the most well-known and widely used formulas in mathematics is the binomial expansion formula, which is:

$(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k$

where ${n \choose k}$ is the binomial coefficient, also known as the number of combinations of $n$ items taken $k$ at a time.

Pascal's Triangle and Binomial Coefficients

Pascal's Triangle is a triangular array of numbers in which each number is the number of combinations of a certain size that can be selected from a set of items. The triangle is named after the French mathematician Blaise Pascal, who studied it in the 17th century.

Here is an example of Pascal's Triangle:

       1
      1 1
     1 2 1
    1 3 3 1
   1 4 6 4 1
  1 5 10 10 5 1
 ...

Each number in the triangle is the sum of the two numbers directly above it. The numbers in each row can be used to calculate the binomial coefficients, which are used in the binomial expansion formula.

Expanding (a-b)^5 Using Pascal's Triangle

Using Pascal's Triangle, we can expand $(a-b)^5$ as follows:

$(a-b)^5 = a^5 - 5a^4b + 10a^3b^2 - 10a^2b^3 + 5ab^4 - b^5$

Here's how we used Pascal's Triangle to get this result:

The coefficients of the expansion are the numbers in the 5th row of Pascal's Triangle: 1, 5, 10, 10, 5, 1.
The powers of $a$ and $b$ are determined by the position of the coefficient in the row. The first coefficient is multiplied by $a^5$, the second coefficient is multiplied by $a^4b$, and so on.

Conclusion

In this article, we have seen how Pascal's Triangle can be used to expand binomials, specifically $(a-b)^5$. The binomial expansion formula and Pascal's Triangle are powerful tools in algebra and combinatorics, and have numerous applications in mathematics, science, and engineering.