Two players P₁ and P₂ play a series of 2n games. Each game can result in either a win or a loss for P₁. The total number of ways in which P₁ can win the series of these games is equal to

P₁ must win atleast n+1 games. Let P₁ win n+r games (r = 1, 2,..., n). Therefore, corresponding number of ways is ${ }^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}+\mathrm{r}}$. 

The total number of ways is

 $\begin{array}{c}\sum _{\mathrm{r}=1}^{\mathrm{n}}{ }^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}+\mathrm{r}}{=}^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}+1}{+}^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}+2}+\cdots {+}^{2\mathrm{n}}{\mathrm{C}}_{2\mathrm{n}}\\ =\frac{2^{2\mathrm{n}}{-}^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}}{2}\end{array}$