# Two Kinds Of Partitions

**Discrete Mathematics**Level 4

Let \( p(n) \) be the *number of partitions* of \( n\). Let \(q(n) \) be the number of partitions of \( 2n \) into exactly \(n \) parts. For example, \(q(3) = 3 \) because
\[
6 = 4+1+1 = 3+2+1 = 2+2+2.
\]
Compute \( p(12)-q(12). \)

**Definition**: A partition of an integer is an expression of the integer as a sum of one or more positive integers, called parts. Two expressions consisting of the same parts written in a different order are considered the same partition ("order does not matter").