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Difference between Open a book on something and an open book

open a book on something—(Bookmakers) take bets on certain contests:

  • There is a good deal of interest in the outcome of this ballot, so I thought it might be fun to open a book on the event.

an open book=used of a person or thing that can be readily examined or understood:

  • He kept to himself personally, but his military life was an open book.

See also: close the book on something / close the books on something.