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I am looking to create  an array of LED's attached to switches. There will be 14 LED's in total. 3 green LED's attached to a single 2-way switch, 8 red LED's attached in pairs to 4 3-way switches, and 3 white LED's attached to 3 more 2-way switches (3 seperate 1 switch:1 LED set ups. Nothing wierd or fancy there). I'd like it all to run off a single power source, light powered would be best, but if that turns out to be too complicated I'll settle for battery powered. My questions are: What kind of power source will I need to use? How is the best way to wire it? and What sort of equipment will I need (resistors, capacitors, etc)?

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After some math and re design I've come up with the following: 2 9V batteries in parallel powering 14 LED's 3 green in series with a 120 Ohm resistor, 8 red in parallel (paired on a 3 way switch) with a 390 Ohm resistor each, and 3 white in parallel with a 330 Ohm resistor each. That sound about right? Everything is run in parallel except the 3 green LED and every thing has a switch, and there's a master switch near the batteries.

I would figure which "line" will use the most power, and use that as a basis for your power supply. Then you can figure, using Ohm's law, what resistors you use depending on the power requirements of each "line" of LEDs. This does not sound like you will need anything other then the PSU, the wire, the LEDs, the switches and the resistors.

Sounds good. I'm guessing the set of 3 green LEDs will be drawing the most power (will verify before proceding). So I get the power requirements of that set and get a power source that will be able to handle it and add a resistor to each line that will bring the totals up to the output of the power supply, correct?

Yes, figure each according to "how many" LEDs, what the power draw of each LED is (different draws for different LEDs), using the power supply you have for the highest draw, in series, I presume.