Notice that $P(n-1)$ counts the number of partition of $n$ that contain $1$, while $P(n-2)$ counts the number of partition of $n$ that contain $2$.

It follows that $P(n-1)+P(n-2)-P(n)$ equals the difference between the number of partitions of $n$ that contain $\{1,2\}$ and the number of partitions on $n$ that contain neither $1$ nor $2$.

Notice that $P(n-1)$ counts the number of partition of $n$ that contain $1$, while $P(n-2)$ counts the number of partition of $n$ that contain $2$.

It follows that $P(n-1)+P(n-2)-P(n)$ equals the difference between the number of partitions of $n$ that contain $\{1,2\}$ and the number of partitions of $n$ that contain neither $1$ nor $2$.