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How many red 5mm leds i can power with a 12 V 1.5 A AC-Adapter?

Welcome to EP!
The number you can drive will depend on how much current each LED draws. Most LEDs like that will take 20mA ok, but you can choose a lower current and hence increase the number of LEDs used.
Yes, you will need some resistors.
Does your mains adapter provide a DC output or an AC output?
Is the output voltage regulated or not?
 
This is how you could use the LEDs, arranged in strings of 5, each string having a resistor.
LED-strings.PNG
Only 2 strings are shown here.
With the resistor value shown, the LED current would be about 17mA per string, so you would be able to have 1.5A/17mA = 88 strings in theory. In practice, allowing for some margin, 80 strings should be manageable, i.e. a total of 400 LEDs.
 
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