Question

Question 3 Consider a team of eleven (11) soccer players, all of whom are equally good players
and can play any position.

(a) Suppose that the team has just finished regulation time for a play-off game and the score
is tied with the other team. The coach has to select five players for penalty kicks to decide
which team wins the game. Since each player takes penalty kicks differently, the order in
which the players are arranged for the penalty kicks is important and can affect the
outcome. How many different ways can the coach select five (5) players to take the
penalty kicks?

(b) A couple of weeks later, the coach wants to form two (2) teams of five (5) from the
eleven players on the team for a scrimmage game (i.e., just a small practice game where
player positions are not important). The eleventh player will act as the referee. How many
ways can the coach divide the team into two teams of five players?

(c) Another week later, the coach wants to test the players to be able to select a captain for
the team. Therefore, again the coach wants to form two (2) teams of five (5) from the
eleven players on the team for a scrimmage game, with the eleventh player again acting
as the referee, but with a small change. The first person chosen for a team of five will be
the captain of the team and will have extra responsibilities. For the rest of the players,
their roles and positions are not important. How many ways can the coach divide the
team into two teams of five players with one captain for each team?

Answer 1

Solution

Back-up Theory

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}……(1)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN……. (1a)

Number of ways of selecting r things out of n things when order of selection is

important is given by: n!/(n - r)! ……………………….........……………………………..…. (2)

Values of n!can be directly obtained using Excel Function: Math & Trig FACT…....…. (2a)

Now to work out the solution,

Part (a)

Vide (2),

Number of different ways the coach can select 5 players out of 11 players to take the
penalty kicks, where the order in which the players are arranged for the penalty kicks is important

= (11!)(11 – 5)!

= 11!/6!

= 39916800/720

= 55440 Answer 1

Part (b)

Vide (1), one team of 5 players can be selected in ¹¹C₅ways

another team of again 5 players can be selected out of the remaining 6 players in ¹⁶C₅ ways and then the left over player is the referee.

Thus,

Number of different ways the coach can select one team of 5 players, another team of again 5 players and the 11^th player as the referee out of 11 players, where the player’s positions are not important

= (¹¹C₅) x (⁶C₅) x 1

= 462 x 6

= 2772 Answer 2

Part (c)

First player selected becomes the captain of the first team. Number of selections possible is

¹¹C₁ = 11.

Once this is done, any 4 out of the remaining 10 players can join the caption to form the first team.

Number of selections possible is ¹⁰C₄ = 210.

Now, from the remaining 6 players, first player selected becomes the captain of the second team. Number of selections possible is ⁶C₁ = 6.

Once this is done, any 4 out of the remaining 5 players can join the caption to form the second team. Number of selections possible is ⁵C₄ = 5.

And then the left over player is the referee. Thus, the total number of selections possible is:

11 x 210 x 6 x5 x 1 = 69300 Answer 3

DONE