# Question 3 Consider a team of

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

(a) Suppose that the team has just finished regulation time for aplay-off game and the scoreis tied with the other team. The coach has to select five playersfor penalty kicks to decidewhich team wins the game. Since each player takes penalty kicksdifferently, the order inwhich the players are arranged for the penalty kicks is importantand can affect theoutcome. How many different ways can the coach select five (5)players to take thepenalty kicks?

(b) A couple of weeks later, the coach wants to form two (2) teamsof five (5) from theeleven players on the team for a scrimmage game (i.e., just a smallpractice game whereplayer positions are not important). The eleventh player will actas the referee. How manyways can the coach divide the team into two teams of fiveplayers?

(c) Another week later, the coach wants to test the players to beable to select a captain forthe team. Therefore, again the coach wants to form two (2) teams offive (5) from theeleven players on the team for a scrimmage game, with the eleventhplayer again actingas the referee, but with a small change. The first person chosenfor a team of five will bethe captain of the team and will have extra responsibilities. Forthe rest of the players,their roles and positions are not important. How many ways can thecoach divide theteam into two teams of five players with one captain for eachteam?

Solution

Back-upTheory

Number of ways of selecting r things out of n things is given bynCr = (n!)/{(r!)(n – r)!}……(1)

Values of nCr can be directly obtainedusing Excel Function: Math & Trig COMBIN……. (1a)

Number of ways of selecting r things out of n things when orderof 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 thesolution,

Part (a)

Vide (2),

Number of different ways the coach can select 5 players out of11 players to take thepenalty kicks, where the order in which the players are arrangedfor the penalty kicks is important

= (11!)(11 – 5)!

= 11!/6!

= 39916800/720

Part (b)

Vide (1), one team of 5 players can be selected in11C5ways

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

Thus,

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

= (11C5) x (6C5) x1

= 462 x 6

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 canjoin the caption to form the first team.

Number of selections possible is 10C4 =210.

Now, from the remaining 6 players, first player selected becomesthe captain of the second team. Number of selections possible is6C1 = 6.

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

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

11 x 210 x 6 x5 x 1 = 69300 Answer3

DONE

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