Question

Why does the sixth row go 1, 6, 15, 20, 15, 6, 1?

Answer 1

There are at least 2 ways of prooving it.

Explanation:

As I wrote you can calculate the elements of Pascal's Triangle in at least 2 ways:

1) Directly from the definition.

Each row consists of numbers: `(""_0^n)`, `(""_1^n)`,..., `(""_n^n)`, where:

`(""_k^n)=(n!)/(k!*(n-k)!`

2) You can construct in graphically:

First row consists of a single number `1`.
Second row consists of 2 numbers `1`
In all other rows first and last numbers are `1`, others are the sum of the 2 numbers in the row above.
The picture shows 10 rows of the triangle.