This book is the first volume and part of a ten-year period of..
This book is the first volume and part of a ten years period of..
This book is the first volume and part of the ten year period of..
Which is correct?
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Number (1) is correct. When a hyphen is used to join two words, they become one "thing" together as a whole. E.g.
Note that the "A" is referring to the number of objects being described, i.e. "period" and "machine". "Ten-year" is an adjective for "period", and "multimillion-dollar" is an adjective for "machine".