Definitely played the odds!
The answer is revealed in the photo of 19 cents, with the arrow pointing at the 17th coin if counted from the top left to right. At first I just photo'd the coins in a small box, but then as you pointed out like a 1:150 shot!
So, I laid the coins out and that's when I had the epiphany to position the 1917 as the "17" coin. I cheated just a wee bit by declaring 6,717 coins sorted when it was actually 6,714 - just to present another clue.
The 19 coins in the hoard: Drum roll please......
2 - 1955 - P
1952 - P
1951 - D
1950 - P
2 - 1948 - P
1946 - P
1945 - P
2 -1945 - D
1944 - P
1944 - S
1941 - P
1940 - P
1939 - D
1934 - P
1935 - P 1917 - P (196,429,785 mintage)