Generally, on the first tee, if everyone is playing from the same tees, order can be determined by flipping a tee in the air and seeing who it points to. Gender isn't really a factor unless you are simply being a gentleman and allowing the lady or ladies to hit first. After playing the first hole, you would simply follow the 'honor' system where the teeing order is determined by scores on the previous hole. Lowest score hits first, high hits last.
However, if the woman or women are in fact playing from the women's tees, the men generally tee off first. This is not gender bias but simply in the interest of faster play and safety. Since the women's tees are further up it makes sense for the men to play first since you pass their teeing area first. Furthermore, if a woman were to tee off first she would be ahead of the men's tees when they began teeing off and risk being hit by a ball.
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