In the last neighborhood football championship where I live, six teams participated, named $A, B, C, D, E$, and $F$. Each team played exactly one match against each of the others. The league table is provided below, where $\mathrm{V}$ is the number of wins, $\mathrm{E}$ the number of draws, $\mathrm{D}$ the number of losses, GP the number of goals scored, and GC the number of goals conceded for each team.

(a) How many matches were played?

(b) The table is incomplete. Determine the number of wins for team $F$, the number of losses for team $D$, and the number of goals scored by team $F$, represented in the table by $x, y$, and $z$.

| Team | $\mathbf{V}$ | $\mathbf{E}$ | $\mathbf{D}$ | GP | GC |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $A$ | 4 | 1 | 0 | 6 | 2 |
| $B$ | 2 | 1 | 2 | 6 | 6 |
| $C$ | 0 | 3 | 2 | 2 | 6 |
| $D$ | 1 | 1 | $y$ | 3 | 6 |
| $E$ | 0 | 1 | 4 | 1 | 5 |
| $F$ | $x$ | 1 | 0 | $z$ | 3 |
(a) Each of the six teams played exactly one match against each of the other five. Therefore, a total of $\frac{1}{2}(6 \times 5)=15$ matches were played.
(b) Each team played exactly 5 matches. Thus, from $x+1+0=5$ it follows that $x=4$. Similarly, for team $\mathrm{D}$ we have $1+1+y=5$, so $y=3$. The total number of goals scored in the tournament is equal to the total number of goals conceded, that is, $6+6+2+3+1+z=2+6+6+6+5+3$, or $18+z=28$, so $z=10$.