(m+n)^6 Pascal's Triangle

Jasonbradley

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Jun 16, 2024

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Expanding (m + n)^6 using Pascal's Triangle

Pascal's Triangle is a powerful tool for expanding binomials raised to a power. It provides the coefficients for each term in the expansion. Let's see how to use it to expand (m + n)^6.

Understanding Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The first few rows are:

           1
          1 1
         1 2 1
        1 3 3 1
       1 4 6 4 1
      1 5 10 10 5 1
     1 6 15 20 15 6 1 

Key Observations:

• Rows: The rows are numbered starting from 0.
• Coefficients: The numbers in each row represent the coefficients in the expansion of (x + y)^n, where 'n' is the row number.
• Symmetry: The triangle is symmetrical, meaning the numbers on the left and right sides are the same.

Expanding (m + n)^6

To expand (m + n)^6, we need the coefficients from the 6th row of Pascal's Triangle: 1 6 15 20 15 6 1.

Now, follow these steps:

1. Powers of m: The powers of 'm' decrease from 6 to 0, starting with m^6.
2. Powers of n: The powers of 'n' increase from 0 to 6, starting with n^0.
3. Coefficients: Multiply each term by the corresponding coefficient from Pascal's Triangle.

Therefore, the expansion is:

(m + n)^6 = 1m^6 + 6m^5n + 15m^4n^2 + 20m^3n^3 + 15m^2n^4 + 6mn^5 + 1n^6

Summary

Using Pascal's Triangle to expand binomials is a straightforward and efficient method. The triangle provides the coefficients, and you simply need to adjust the powers of the variables.