A world series is a best of $7$ series between team $A$ and team $B.$ It takes $4$ wins to win the series. How many ways can a team win the World Series?
Suppose that a World Series is completed leading to an extended series where extra exhibition games are played (between the same two teams $A$ and $B$) after the series has already been won. These exhibition games will always be won by the team that lost the official World Series and are to continue until that losing team also has a total of $4$ wins. In this way, if the results of the World Series read $ABAABA$, then the results of the extended series read $ABAABABB$.
The sequence of results for the extended series is just an arrangement of 4 copies of the letter A and $4$ copies of the letter B. There are $_8C_4$ = $8*7*6*5/(4*3*2*1) = 70$ such arrangements. (We may choose $4$ positions for the letter $A$ from $8$ possible positions.)
Each sequence of results for the extended series determines exactly one possible sequence of results for the World Series (by omitting the exhibition games).
I am not sure if this is correct. Is there a simpler way of doing this? Can someone show me?