Standpipes and laundry sink

[Patrick]

I'm going to replace an old concrete double width laundry tub with a newer plastic setup. The existing has no standpipes. Instead, the washing machine is hooked over one end of the tub and the opposite side of the tub has the water softener drain. Since these drains are a good 5 ft. apart I'm strongly considering putting in a new sink with a separate standpipe on each end of the tub, one for the washing machine and one for the water softener. If I did that, should I have 3 p-traps, one on each standpipe and one on the sink? Any other thoughts on how I plan to do this? I am aware that I need an air gap on the standpipe for the water softener.

Thanks for any guidance you can give me.

 

[Patrick]

One other thing on this. The existing tub and new tub are both within a few feet of the main stack that drains out of the house, which is of course vented. These two standpipes and the laundry sink will converge into one 2" drain pipe that goes right into that stack. So, I'm assuming that no other venting is required. Do you think that's a safe assumption?