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There are two general ways of bowling a 289 game:
1) Start with 10 strikes, then 9 count, and pick up the spare
2) Start with a spare, then 10 strikes, then a 9 on your last ball.
Within those general ways, there are many variations.
In the first, there are theoretically 10 different ways to get a 9-count on the 11th ball.
In the second, there are theoretically 1,023 ways to get a spare in the first frame, and 10 different ways to get a 9-count on the last ball.
In tenpin bowling, it's pretty hard as the bowler must throw 10 strikes in a row.
The only two ways I know of a tenpin bowler getting a 289 score is ...
1) Get a spare in the 1st frame, then throw 10 strikes in a row (from the 2nd frame through the second ball in the 10th frame), and then get a 9 count on the last ball in the tenth frame.
2) Throw 10 strikes in a row (1st frame through the first ball of the 10th frame), then get a 9 count with the second ball in the 10th frame and convert the spare with the third ball.
To score a 279 in 10 Pin Bowling, you would need to bowl a Spare in the first frame followed by 8 strikes & for the tenth frame you would need a Strike followed by a 9 spare as shown below.
9/, X, X, X, X, X, X, X, X, X, 9/
The score would show as this during the game
20,50,80,110,140,170,200,230,259,279