Interpretation of Backtracking Algorithm Problem - weighted matching
Suppose man $i$ is paired with woman $w(i)$, for $1\le i\le n$. 

"The sum of the products of the preferences" is the following number.

$$ P_{1,w(1)}Q_{w(1), 1} + P_{2,w(2)}Q_{w(2), 2} + \cdots + P_{n,w(n)}Q_{w(n), n} $$ 

One can imagine that each individual product $P_{i,w(i)}Q_{w(i),i}$ quantifies how preferred is the pairing between man $i$ and woman $w(i)$.