## Picking a Football Team

A football squad consists of twenty two players, from which a team of eleven is selected, and three substitutes.How many selections are possible. Typically, a goalkeeper can only play in goal, while an outfield player may play in more than one position. Suppose that the squad contains three goalkeeper. A goalkeeper must be picked for the team, and there must also be a goalkeeper picked as substitute. There are three ways to pick the team goalkeeper and then two ways to pick the substitute, so

\[3 \times 2 =6\]

ways to pick the goalkeepers.There are 10 outfield players remaining to be picked from a choice of 19. Suppose for simplicity that any of these players can play in any position. Each player is to be assigned to a particular position. All the positions are distinct. We can pick 10 players from a possible 19 in

\[{}^19 C_10 =\frac{19!}{10! 9!}\]

ways, and then assign these players to their positions in \[10!\]

ways, so there are \[{}^19 C_{10}\times 10! = \frac{19!}{10! 9!} \times 10! = \frac{19!}{9!}\]

of picking the outfield team.There are

\[6 \times \frac{19!}{9!}\]

ways of picking the team.