Pascal's Triangle: A Wonder Of Mathmetics

Pascal's triangle is a wonderful element of math. It can help us solve numerous math problems! Pascal's triangle consists of "rows" and "diagonals". Let me show you what this means:

Diagonal 1: 1-1-1...

Diagonal 2: 1-2-3...

Diagonal 3: 1-3-6...

Diagonal 4: 1-4...

Row 0: 1

Row 1: 1

1Row 2: 1 2 1

Row 3: 1 3 3 1

Row 4: 1 4 6 4 1

Do you notice anything? The triangle begins with 1s on the sides and keeps adding ones on the left and right edges. For the numbers that aren't on the sides, each number is the sum of the two numbers above it.

Here is a formula for the nth number in diagonals 1, 2, 3, and 4:

Diagonal 1: All 1's

Diagonal 2: n, because these are the counting numbers

Diagonal 3: These are the triangle numbers. Let me quickly show you how to calculate the nth triangle number.

Say n = 2. We have the 2nd triangle number, which is 1 + 2.

We can represent this with a triangle:

xxx

(x = 1 dot)

If we copy the triangle and make it so that the diagonals align, we get a rectangle.

We have one side with length 2, and one side with 2 + 1, so the area is 2 × (2 + 1). But we used 2 triangles, so we divide by 2:

2 × (2 + 1) / 2 = 3

But what if we had, say, the 10th triangle number?

We could replace 2 with 10:10 + 1 = 1111 × 10 / 2 = 55

Diagonal 4: This is a bit more complex, but we can get the nth number of the diagonal by using the formula "n choose 3" (n! / (3! × (n − 3)!)).

General formula:

The mth number on the nth diagonal will be "m choose (n − 1)" (m! / ((n − 1)! × (m − n + 1)!)).

(By the way, anything choose 0 = 1.)

Thanks for reading!