Federated Learning (FL) is an emerging paradigm in machine learning without exposing clients' raw data. In practical scenarios with numerous clients, encouraging fair and efficient client participation in federated learning is of utmost importance, which is also challenging given the heterogeneity in data distribution and device properties. Existing works have proposed different client-selection methods that consider fairness; however, they fail to select clients with high utilities while simultaneously achieving fair accuracy levels. In this paper, we propose a fair client-selection approach that unlocks threefold fairness in federated learning. In addition to having a fair client-selection strategy, we enforce an equitable number of rounds for client participation and ensure a fair accuracy distribution over the clients. The experimental results demonstrate that FedFair^3, in comparison to the state-of-the-art baselines, achieves 18.15% less accuracy variance on the IID data and 54.78% on the non-IID data, without decreasing the global accuracy. Furthermore, it shows 24.36% less wall-clock training time on average.