In this paper, we study the power allocation (PA) scheme design for energy efficiency (EE) maximization with discrete-rate adaptive modulation (AM) in the downlink distributed antenna system (DAS). By means of the Karush-Kuhn-Tucker (KKT) conditions, an optimal PA scheme with closed-form expression is derived for maximizing the EE subject to maximum transmit power and target bit error rate (BER) constraints, where the number of active transmit antennas is also derived for attaining PA coefficients. Considering that the optimal scheme needs to calculate the PA of all transmit antennas for each modulation mode, its complexity is extremely high. For this reason, a low-complexity suboptimal PA is also presented based on the antenna selection method. By choosing one or two remote antennas, the suboptimal scheme offers lower complexity than the optimal one, and has almost the same EE performance as the latter. Besides, the outage probability is derived in a performance evaluation. Computer simulation shows that the developed optimal scheme can achieve the same EE as the exhaustive search based approach, which has much higher complexity, and the suboptimal scheme almost matches the EE of the optimal one as well. The suboptimal scheme with two-antenna selection is particularly effective in terms of balancing performance and complexity. Moreover, the derived outage probability is in good agreement with the corresponding simulation.