Pascal's triangle

Pascal's triangle is a triangle that works in the following way.

The sum of the numbers in each row is 2 to the nth power. (Remember, the first row is row zero.)

Properties

Except for the first column, if one alternates the sum and difference (difference first), the value is always 0.

For example,

Row 2: 1 - 1 = 0

Row 3: 1 - 2 + 1 = 0

Row 4: 1 - 3 + 3 -1 = 0

Combinatorics approach

The triangle can also be viewed as follows:

This can be used to prove the identity that

$\sum _{k=0}^{n}{\binom {n}{k}}=2^{n}$

The $n$-th row of the triangle, starting with zeroth row, represents the coefficients of the binomial expansion $(a+b)^{n}$ .