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....

row 0 1. diagonal 2 1-2-3....

row 1 1 1 diagonal 3 1-3-6...

row 2 1 2 1 diagonal 4: 1-4...

row 3 1 3 3 1 and so on..

row 4 1 4 6. 4 1

Do you notice anything?the triangle begins with 1 1-1, and keeps adding ones on the left and right sides, and for the ones that aren't on that side, that is the sum of the 2 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 2rd triangle number, which is 1+2. We can represent this with a triangle:

x

xx

(x=1 dot)

If we copy the triangle and make it so that the diagonals align, we get this:

xyy

xxy

We get a rectangle!

we have one side with length 2, and one with 2+1, so the area is 2(2+1)! But we used 2 triangles so we get

2(2+1)/2! Because we had our 2nd triangle number, we get 3. But what if we had, say, the 10th?

We could replace 2 with 10! 10+1=11, 11*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)!) (BTW anything choose 0=1)

That'll be it for now! Thanks for reading!