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1) At the height of its popularity in the 1960s, the designer Emilio Pucci helped to create the "mod" fashion craze, a dramatic reaction to the traditional tailored look in women's wear.
###Height of popularity modifies Emilio. Incorrect.
2) At the height of his popularity in the 1960s, the "mod" fashion craze, by designer Emilio Pucci, was a dramatic reaction to the tradional tailored look in women's wear
###Again, error in modifier.
3)At the height of its popularity, the "mod" fashion craze was created partially by designer Emilio Pucci, reacting dramatically to the traditional tailored look in women's wear.
###How can the fashion be at the height of its popular even before it was created?
4)Reacting dramtically to the traditional tailored look in women's wear, designer Emilio Pucci helped to create the "mod" fashion craze at the height of its popularity in the 1960s.
###How can the fashion be at the height of its popular even before it was created?
5)Designer Emilio Pucci helped to create the "mod" fashion craze that reached the height of its popularity in the 1960s and was a dramatic reaction to the traditional tailored look in women's wear.
###Wordy but correct.