Two team playoff question, homework help I am working through my college text book (mathmatical statistics freund/walpole) trying to refresh my stat skills. Its been a couple years...
I would sure appreciate any pointers on an exercise question that has me stumped...
In a two -team playoff in some sport, the winner is the first team to win m games.
(a) Counting separately the number of play offs requiring m,m +1,....,  2m-1 games, show that the  toatl number of different outcomes (sequences of wins and losses) is 
$$
2\left[ {{m-1}\choose {m-1}}  +  {{m}\choose {m-1}} +   ... + {{2m-1}\choose{m-1}}\right]
$$
(end of question)
I drew this out as a path problem, wins along the X and wins losses along the Y. By inspection I can see that one team can win a best of 5 (m=3) in the following ways 
Team A has to win 3 games, but they can only loose 2 in the process...
$$
\left[ {{3}\choose {3}}  +  {{4}\choose {3}} +  {{5}\choose{3}}\right]
$$
The other team might win the series as well, so I would need to multiply the above by 2.
I can generalize this to 
$$
2\left[ {{m}\choose {m}}  +  {{m+1}\choose {m}} +   ... + {{2m-1}\choose{m}}\right]
$$
I have tried various substitutions but I can not seem to get to the form asked for. 
I notice they are using (m-1) in the bottom. For my working scenario, this would equate to 
$$
2\left[ {{2}\choose {2}}  +  {{3}\choose {2}} + {{4}\choose {2}} + {{5}\choose{2}}\right]
$$
This only makes sense to me in the case of a full series, or a sweep since 
$$
{{5}\choose{3}} = {{5}\choose{2}}   and   {{3}\choose{3}} = {{2}\choose{2}} 
$$
Are they counting the ways for Team B to loose? 
Any tips or pointers? I am sure I am missing something fundamental.
Thanks in advance.

For the play-off requiring $m$ games, the number of possible sequences of wins is given by:
$$n_m={m\choose{m}}={{m-1}\choose{m-1}}$$
For the play-off requiring $m+1$ games, the number of possible sequences of wins is given by:
$$n_{m+1}={{m+1}\choose{m}}={{m}\choose{m}}+{{m}\choose{m-1}}={{m-1}\choose{m-1}}+{{m}\choose{m-1}}$$
...
For the play-off requiring $2m-1$ games, the number of possible sequences of wins is given by:
$$n_{2m-1}={{2m-1}\choose{m}}={{2m-2}\choose{m}}+{{2m-2}\choose{m-1}}\\=\left[{{m-1}\choose{m-1}}+{{m}\choose{m-1}}+...+{{2m-3}\choose{m-1}}\right]+{{2m-2}\choose{m-1}}$$
Using the same argument you used, we consider the possible sequences of losses (the other team wins) and that gives:
$$n_{Total}=2\left[{{m-1}\choose{m-1}}+{{m}\choose{m-1}}+...+{{2m-2}\choose{m-1}} \right]$$
