I want you to teach me, explain, and give examples of pascal triangle

Here's how it's constructed:

1. The top row (row 0) starts with a single 1.
2. Each subsequent row starts and ends with a 1.
3. Every other number in a row is the sum of the two numbers directly above it in the previous row.

Here's the beginning of the Pascal triangle:

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

Example of construction:

• In Row 3, the '3' is the sum of the '1' and '2' above it.
• In Row 4, the '6' is the sum of the '3' and '3' above it.

The Pascal triangle has many interesting properties and applications, particularly in algebra (for binomial expansions) and probability.

For instance, the numbers in the nth row of the Pascal triangle (starting with n=0) are the binomial coefficients for the expansion of (x + y)ⁿ.

Example of binomial expansion:
To expand (x + y)³, we look at Row 3 of the Pascal triangle: 1 3 3 1.
So, (x + y)³ = 1x³y⁰ + 3x²y¹ + 3x¹y² + 1x⁰y³
Which simplifies to: x³ + 3x²y + 3xy² + y³