Discussion:  

  If you consider checkerboard coloring, all pieces other than C
  contain two squares of each color.  Therefore, any puzzle with an
  odd number of C pieces collectively is certain to be unsolvable.
  In our examples and tests, we make sure to rely on unsolvable
  puzzles with an even number of C pieces.


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Public Examples
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Input: BCC  BDE

#####.     CCCBB   EEDDD
.##.#.      Cc B    EE D
..###.       ccB     BBB
..#...       c       B
......
......

Note:      This is the primary example discussed in the prose.
           Demonstrates two copies of same piece of same color (but
           different rotations), and two copies of sampe peice, one in
           each color (again with different rotations).  Also
           demonstrates that there could be a hole in the resulting shape.
Pieces:    BCDE
Rotations: B(0), B(90), C(0), C(270), D(0), E(0)

============================================

Input: BCC EFG

Impossible  (slight variation from first case)

============================================

Input: AAA CCE

####..  AAAA    CCCE
.####.   aaaa    CEEc 
..####    AAAA    Eccc
......
......
......

============================================

Input: ADG DEE

####..   AAAA    EEee
.####.    DDDG    EEee
...###      DGG    DDD
.....#        G      D
......
......

Note:      Just a fun shape.  Uses all six columns, but never in the same row.
Pieces:    ADEG
Rotations: A(0), D(0), E(0), G(90)


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Judge Tests
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Input: AAD CCF

##....    DD    FF
##....    DA    FF
###...    DAa   CCC
.##...     Aa    Cc
.##...     Aa    cc
..#...      a     c

Note: uses all six rows
Pieces:     ACDF
Rotations:  A(90), D(270), F(*), C(0), C(90)

============================================

Input: BCC GGG

.####.     CBBC    1122  
######    CCCBCC  112233 
...##.       BC	     33  
......
......
......

Note: top-left corner is left empty.
      Uses all six columns in second row.
      Blue is a triple of same piece,
      and ends up laid out without any rotaiton.
      Red has a double, with one rotated and one not.
Pieces:     BCG
Rotations:  B(90), C(180), C(270), G(0)

============================================

Input: CFG CCC

Impossible

============================================

Input: ADF BEG

####..    AAAA   BBBG
#####.    DDDFF  BEEGG
..###.      DFF    EEG
......
......
......

Note:       Uses six distinct pieces.
Pieces:     ABDEFG
Rotations:  A(0), D(0), F(0),  B(0), E(0), G(90)

============================================

Input: AEE BDG

###...   AEE    BBB
#.##..   A EE   B DD
#.#...   A e    G D
###...   Aee    GGD
.#....    e      G
......


Pieces:    ABDEG
Rotations: A(90), B(0), D(90), E(0), e(90), G(90)