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how many random words...........

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I think there's some room for interpretation on this.  I'm going on the assumption that you must use those keys in that order (i.e. get a letter from the 2 key, a letter from the 3 key, etc.).  I'm also assuming it can be jibberish or the question would be really unfair. :-)
If that's correct, this in essence, this is not really different than the dice problem.  It only seems different.  But if you always think of these types of problems as a series of events, and a number of outcomes for each event, they actually do work the same.
If you press 2, how many outcomes can you have with the letters.  That is, how many possibilities?  (A B or C, so 3).  So that's your first event, hitting the 2 key.
The 2nd event is hitting the 3 key.  How many outcomes of letters for the 3?  Well, 3 again because there's 3 letters on there.
The 3rd event is hitting the 5 key.  How many outcomes?
Then you hit the 7 key.  How many outcomes?
Then you  hit the 9 key.  How many outcomes?  (Careful on that one.)
So once you know how many outcomes for each event you have, you can again use your multiplication rule --> just multiple the outcomes of each event to get the final answer.
One hint on doing these: distinguish between "and" and "or."  You have 5 events, so it's key 2 AND key 3 AND key 5, etc.  The "or" is the outcomes.  Key 2 has 3 outcomes - A OR B OR C.  If you write the "and"s across the top in a row, and then put the number of outcomes right below each, it's those outcomes you're going to be multiplying.  Don't ever multiply outcomes times events.