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the average lifespan of a major league ball is 9 pitches
the average number of pitches thrown in a game ~ 140
162 games in a season
30 teams
(140*162*30)/9 = 75,600
EDIT:
The correct numbers are 143 pitches per team on average which is 286 per game. According to MLB they use 65 balls per game. 30 teams have 162 games, but each team has to face another team so the number would be 15*162 which yields 2430. Multiply 2430 by 65 and you get 157,950. There's your number!
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