The people who picked Begich as their first choice had their second choice counted, but the people who picked Palin as their first choice threw away their second choice. Had the latter's second choices not been thrown away, then Begich would have won.
> The people who picked Palin as their first choice threw away their second choice.
You don't get to rank a bunch of choices and then claim your vote didn't get counted because every one of your choices didn't win the election. People who picked Palin as their #1 choice won against Begich. Their vote went to Palin exactly as they wanted. Palin lost, but the voter who put Palin first voted for the person they wanted and that vote was counted.
Voting for a losing candidate does not mean throwing your vote away or not having your vote counted.
People who ranked Begich second still had the vote for their #1 pick counted. It doesn't matter who your #2 pick was if your number #1 pick is still in the race. That's the entire point of picking a #1 candidate.
A vote that is counted in the final contest of an election is not a vote "thrown away" even though the person they voted for lost. Voters have to be allowed to pick a candidate who ultimately doesn't have enough supporters to win an election. That's how elections work.
