THE SOLUTION:

The first song was by Bangles, vote followed by the Beatles, the Bee Gees, Blondie, and the Beach Boys.

Here’s a sample solving sequence for this puzzle. (Your manner of solving may have been different):

Clue #1 reveals that:

• The Beach boys weren’t first or second.
• Blondie wasn’t first or last.
• The Beatles weren’t fourth or fifth.

Clue #2 reveals that:

• The Bangles weren’t fourth or fifth.
• The Bee Gees weren’t first or second.

From there:

• The Bangles could not have been second without contradicting Rule #1, so the Bee Gees could not have been fourth.
• The Bangles could not have been third without contradicting Rule #1, so (by elimination) the Bangles were first and the Bee Gees third.
• Then, going back to Clue #1, Blondie has to be fourth and the Beach Boys fifth,
  leaving the Beatles as second.

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