There were 17 official printings of this book, though printings 6 to 10 all say "6th printing".
In the 1970s, small publishers (such as those used by TSR) produced hardback books in a multi-step process involving human labor at every step. Pages were produced in batches with an attempt to have each page's count within about 5% of the average page count. Folding and stitching pages into 16-page "signatures" were additional steps. Attaching endpapers and trimming were additional steps. Producing the hardback cover was a separate process. Whichever portion of the book assembly ran out first determined the total size of the print run.
TSR, especially in the early printings, was particularly frugal when producing hardback materials; parts were not wasted and leftover pieces of books from one Printing subsequently were used early on in the next Printing. Expediency won out over consistency. This resulted in numerous "hybrid" editions, where parts of a previous printing were used to "fill in" subsequent Printings. As TSR got bigger (and more professional) things like this happened less and less often. Later printings are much more consistent throughout.
One of the easiest "hybrid" types comes from the late Third Printing books that have some of their 16-page signatures filled in with those from the early 4th printing - the different page shading is apparent (it is unknown how many of these were made).
The Player's Handbook has enjoyed numerous Printings; these usually are easily determined by information on the copyright page, though interested parties should consult expert resources when purchasing or investigating specific, early printings. Note that most of the corrections mentioned as "errata" in Dragon (Issue 35 - Mar 1980) never were incorporated into any Printing.
The 1st - 8th Printings feature the "old" cover of thieves stealing the gem-eye of a large demonic idol; the 9th and later Printings feature the "new" cover of a wizard casting a spell.
1st Printing - June 1978 (some collectors identify early and later versions); wizard TSR logo
2nd Printing - "2nd Printing, September 1978" on copyright page; wizard TSR logo
3rd Printing - "3rd Printing, January 1979" on copyright page; wizard TSR logo
4th Printing - "4th Printing, May 1979" on copyright page; wizard TSR logo
5th Printing - "5th Printing, October 1979" on copyright page; wizard TSR logo
6th Printing - "6th Printing, January 1980" on copyright page; endpaper are orange; ISBN on spine; wizard TSR logo
7th Printing - "6th Printing, January 1980" on copyright page; endpapers are white; ISBN on spine; wizard TSR logo
8th Printing - "6th Printing, January 1980" on copyright page; endpapers are white; ISBN and Product Number on spine; face TSR logo
The 9th and later Printings have an orange spine and revised cover art of a wizard casting a spell.
9th Printing - "6th Printing, January 1980" on coppyright page; updated cover art with orange spine - early versions incorrectly credit the cover to D. A. Trampier, while later versions correctly credit Jeff Easley; ISBN and Product Number relocated to back cover; angled TSR logo
10th Printing - "6th Printing, January 1980" on copyright page; UPC bar code added on back cover - early versions do not have a price, while later versions add the $15.00 price near the UPC bar code; angled TSR logo
The 11th - 17th Printings correctly identify the Printing and date on the copyright page.
11th Printing - August 1987
12th Printing - November 1987
14th Printing - October 1988
15th Printing - July 1989
17th Printing - July 1990 (a year after the 2nd Ed. PH had been released)
Subsequently, "premium" reprints of the book have been made available though they are all clearly marked at such.
Exhaustive information, including a diagnostic flowchart, can be found at https://www.acaeum.com/ddindexes/setpages/phb.html