Broadband solid state light sources include remote phosphor LED(s), short-wavelength direct emitting LED(s), and long-wavelength direct emitting LED(s). A diffuse or clear cover member covers these LEDs. Each remote phosphor LED includes an LED, a phosphor layer, and a dichroic reflector. The light sources can provide a broadband output light over a wide color range, and can do so efficiently while energizing a high percentage or proportion of the total number of LEDs in the system. The broadband output may for example exhibit a color difference of at least 0.2 in CIE chromaticity units, and/or a correlated color temperature difference of at least 4000 or 5000 Kelvin, while energizing more than half, or at least 60%, or at least 70%, of the total number of LEDs. Numbers of LEDs can be replaced with effective numbers of LEDs if LEDs of substantially different sizes are included in the light source.