The coefficients of the binomial expansion follow a special pattern called Pascal's triangle. Explain this pattern.

Pascal's triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. The first row of the triangle contains only the number 1, and each subsequent row is constructed by adding the two numbers above it.

When expanding a binomial expression, the coefficients of each term in the expansion can be found by looking at the corresponding row of Pascal's triangle. For example, when expanding (a + b)^3, the coefficients are 1, 3, 3, 1, which correspond to the fourth row of Pascal's triangle.

This pattern arises because each term in the expansion corresponds to a specific combination of choosing a certain number of terms from each of the binomial factors. The coefficients in Pascal's triangle represent the number of ways to choose a certain number of items from a set, which is why they can be used to determine the coefficients in the binomial expansion.

Pascal's Triangle Pattern:

Pascal's triangle is an arrangement of numbers in a triangular shape, where the numbers are the coefficients of the binomial expansion. The nth row of the triangle contains the coefficients of the binomial expansion for (x + y)^n.

Construction:

1. Starting Point: The first row consists of a single 1.
2. Recursive Rule: Each subsequent row is constructed by adding the two adjacent numbers in the previous row for the corresponding position.
3. Ends: The ends of each row consist of a single 1.

Pattern:

The pattern of Pascal's triangle is such that the numbers in each row form a binomial coefficient, which is calculated as follows:

```
C(n, r) = n! / (r! - (n - r)!)
```

where:

- C(n, r) is the binomial coefficient
- n is the row number
- r is the column number

Example:

Suppose we want to calculate the 4th row of Pascal's triangle. According to Pascal's triangle, the coefficients for (x + y)^4 are:

```
1 4 6 4 1
```

Binomial Expansion:

Using these coefficients, we can expand the binomial expression as follows:

```
(x + y)^4 = 1 - x^4 - y^0 + 4 - x^3 - y^1 + 6 - x^2 - y^2 + 4 - x^1 - y^3 + 1 - x^0 - y^4
```

This expanded expression represents the terms and their respective coefficients for the binomial expansion of (x + y)^4.

Significance:

Pascal's triangle is a powerful tool in mathematics and computer science. It is used for calculations in probability, statistics, and combinations and permutations. The pattern of Pascal's triangle provides a systematic and efficient way to calculate binomial coefficients and expand expressions quickly and accurately.