The regex "or" (Alternation, it's called) has fairly low precedence. That means that the ^ binds more closely than the |. The result is that you've got this going on:

m/
  ^bing |
   bong |
   bang\d\d
/x;
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I used an "extended regular expression" so that I could group each subexpression (each alternate) on its own line. If you want the ^ to bind to all three, and the \d\d to bind to all three, you must use parenthesis to constrain the alternation. And if you aren't trying to capture, use non-capturing parens:

m/^b(?:i|o|a)ng\d\d/;
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(Note: I factored out everything that is common to all three alternates. That step is unnecessary. You could use (?:bing|bong|bang) too.)

Alternation may be the best route to follow. But sometimes when you see it factored down as the previous example, you might suddenly realize, hey, I can do this with a character class too:

m/^b[ioa]ng\d\d/;
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