We propose a new approach in obtaining the partition function of a system of several interacting particles (fermions) in external field within path integral Monte Carlo method based on direct averaging of the exchange contributions over the positive weight determined by non-closed trajectories. This procedure is performed at fixed temperatures, ranging to rather low values. The complete partition function for each temperature is then obtained within an expanded ensemble procedure. We found a good agreement of data obtained by the new method with results of previously proposed path-integral-expanded-ensemble Monte Carlo calculations [1, 2]. The new approach allows to reach observably lower temperatures compared to method [1, 2], thus providing a significant reduce of the sign problem. It yields an independent way to treat thermal properties of quantum systems, so the good agreement with previous data [1, 2] allows us to test and state the validity of both approaches. Simulations for systems of 2 up to 5 non-in