In smart homes, mining frequent/correct activities' sequences, AS, of specific behaviour, plays a vital role in building smart monitoring systems analyzing daily life activities (DLA), from which, the system can identify anomalies and automatically send alerts to users to remember them regarding any missing activity. Some researchers developed an intelligent system based on the Apriori algorithm, where all frequent k-Activities' sets mined by Apriori are used to identify all their permutations, which are then filtered out to extract just the frequent/correct k-Activities' sequences. However, because of using the Apriori algorithm, this system suffers from repeatedly scanning the DLA dataset and generating a huge number of candidates. As well as the exponential complexity of finding all permutations of all frequent k-activities' set to find the frequent k-activities' sequences〖AS〗^k. In this paper, a new Positional Representation-based Frequent 〖AS〗^k Mining algorithm, PR-FASM has been proposed, which is based on a new representation called Positional Representation (PR) of each activities' sequence of a specific behaviour. PR reflects the correct orders of each 〖AS〗^k across all possible AS of a specific behaviour. PR-FASM overcomes the drawbacks of the mentioned system by scanning the DLA dataset only once and reducing the search space and time for finding the frequent 〖AS〗^k. On a CHESS dataset and a real smart home dataset called CASAS, the experimental results show that the system that is based on the PR-FASM algorithm is more efficient and scalable than the systems based on the Apriori algorithm and other sequence mining algorithms.