Okada and Stembridge's Pfaffian formula for the enumeration of families of nonintersecting paths with fixed starting points and unfixed ending points has been widely used to resolve many challenging problems in enumerative combinatorics. In this talk, we present a new formula that complements Okada and Stembridge's Pfaffian formula. The combinatorial interpretation of the new formula gives a reflection principle for nonintersecting paths. It implies that the enumeration of families of nonintersecting paths with unfixed ending points can be resolved by enumerating families of nonintersecting paths with fixed ending points instead. Using this formula, we also show that the enumeration of lozenge tilings of a large family of regions with free boundaries can be deduced from those without free boundaries and present several applications of this result.