24pin power supply into 20pin motherboard

Hello,
My computer has a Biostar P4M80-M4 motherboard. I am replacing the damaged 20pin 550w ATX power supply. They sent me a new 24pin 600w replacement. I notice that the 24pin connector is an assembly consisting of a 20pin part plus a snap-on 4pin part that can be detached from the 20pin. I suppose this 4pin component is detached to allow insertion of the 20pin connector into the 20 pin motherboard power connector. All the wires on both 20 pin connectors (old 20pin connector and 20pin component of the new power supply) are same colors on same pins, except that on the new supply there is no white cable ( on pin #18 and there are a red and a pink cable on pin #20). Can anybody tell me if this power supply will work o.k. just plugging-in the 20 pin component, do I have to modify the connections or use an adapter?

The 20+4 are usually completely compatible with the older 20-pin motherboards.

Assuming when you are naming the pins you are only naming the ones on the 20 pin segment:

Pin 18 is a no-connection pin, so there is no problem if a cable does not run here.

Pin 20 is a +5v pin, and should be red in color. If it is red and pink, this is likely a coloring defect in the plastic - if this is true, you probably have a red and pink mottled color.

If I am incorrect in any of my assumptions, let me know, and we'll go from there.

You will be fine, just attach that 20pin from PSU to the board, don't worry about that.
Since your board only need 20pin then don't bother with the extra 4pin, the newer board use 24pin and the older board has 20pin.