General Combination Questions

Apr 2010
26
0
just some questions that i need help with, if someone could help, it would be appreciated very much

q1. find the number of ways of arranging 7 people in a line with the tallest in the middle

q2. a squad of footballers has 10 midfielders, 3 goalkeepers and 7 defenders. how many teams (of 11 players) could be selected if the team only has one goalkeeper and at least 5 midfielders and at least 4 defenders?
 

Soroban

MHF Hall of Honor
May 2006
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Lexington, MA (USA)
Hello, gbooker!

(1) Find the number of ways of arranging 7 people in a line
with the tallest in the middle.

Place the lallest in the middle.

Then the other six people can be arranged in \(\displaystyle 6! = 720\) ways.



(2) A squad of footballers has 3 Goalkeepers, 10 Midfielders, and 7 Defenders.
How many teams (of 11 players) could be selected if the team has:
one Goalkeeper and at least 5 Midfielders and at least 4 Defenders?
There are two possible team compositions:

. . \(\displaystyle 1G,\;6M,\;4D\!:\;\;{3\choose1}{10\choose6}{7\choose4} \;=\;22,\!050\) possible teams.

. . \(\displaystyle 1G,\;5M,\;5D\!:\;\;{3\choose1}{10\choose5}{7\choose5} \;=\;15,\!876\) possible teams.


Therefore, there are: .\(\displaystyle 22,\!050 + 15,\!876\;=\;37,\!926\) possible teams.