Consider a series of 7 matches between two teams. The first team which wins 4 matches is the winner. We denote by $p$ the probability that team 1 wins a match and by $q= 1-p$ that team 2 wins a match. We assume that theses probabilities don't change during the matches and that the result of the matches are independent.
How can I show that the probability that the game ends at $k$ matches is given by
$ \left( \begin{array}{llll} k -1 \\ \ \ \ 3 \\ \end{array} \right) \cdot ( p^4 q^{j-4} + q^4 p^{j-4} ), j=4,5,6,7$
Everything is clear to me, just not how to derive that the binomial coefficient
$ \left( \begin{array}{llll} k -1 \\ \ \ \ 3 \\ \end{array} \right)$
is exactly given like that.
If the series ends at $k$ matches, the winning team has to win the last match. $\binom{k-1}3$ is the number of ways to choose the matches before the last match that the series-winning team won.