According to fzabkar in Seagate: Forums: 2TB ST2000DM001 Barracuda from Seagate seriously cripled through redesign in 3/2012 or 4/2012, the S/N (serial number) on the label of your Seagate Desktop HDD ST2000DM001 contains information about the number of plates.
Check the third character of the S/N, F is 3 while E is 2 platters.
The first character of the S/N is the manufacture of head stack components.
S – SU – Suzhou in China
W – WU – Wuxi in China
Z – TK – Teparuk in Thailand