In the World Series, two teams play each other until one team wins 4 games. How many different series of games is possible? (i.e., Team A wins the series by winning games 1, 2, 4, and 6)
Would it just be 7 choose 4?
If Team A wins in 4 games, they must win the first three games. If Team A wins in 5 games, they must win 3 out of the first 4 games. If Team A wins in 6 games, they must win 3 out of first 5 games. And if team A wins in 7 games, they must win 3 out of the first 6 games.
So number of ways Team A wins is
And the number of ways Team B can win is obviously the same.
Team A wins if any of the sequences occurs?
Suppose that we require that seven games be played period no matter the outcomes. Do you see that any string of four A's and three B's mans that team A wins?
As I said above, this question has a long history.
I can trace it back ot at least the century.