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

Question

Serenity Jones · Accepted Answer

There are at least 2 ways of prooving it.

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.

Anna Allen · Answer

undefined The sixth row in Pascal's Triangle represents the coefficients of the sixth-degree polynomial when expanded from $ (a + b)^5 $. Each number in the row corresponds to a term in the expansion, and the values are obtained using combinations (binomial coefficients). Specifically, the numbers in the row correspond to the binomial coefficients of the form $ \binom{5}{k} $, where $ k $ ranges from 0 to 5. These coefficients are symmetric because of the symmetry property of Pascal's Triangle, hence why the row appears as 1, 6, 15, 20, 15, 6, 1.