Fantasy Cricket

For discussing Olympiad Level Combinatorics problems
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Fahim Shahriar
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Fantasy Cricket

Unread post by Fahim Shahriar » Fri Sep 14, 2012 4:09 pm

You are playing a game on T20 World Cup. There are $12$ teams and each team has $5$ Batsmen, $5$ Bowlers, $4$ All-rounders and $1$ wicket keeper in their squad.
You have to create a team of $12$ players; consists of $4$ Batsmen, $3$ Bowlers, $3$ A.R , $1$ W.K and a $12^{th}$ man.[The 12^{th} man can be any of the remaining player]. The condition is- you can select only one player from each team.

How many different teams can be created ?

I made this problem. But I'm not sure about the answer.:?
This problem can be made tougher changing the condition. But it already seems TOUGH! :!: !
Name: Fahim Shahriar Shakkhor
Notre Dame College

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nafistiham
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Re: Fantasy Cricket

Unread post by nafistiham » Sun Sep 16, 2012 11:38 am

I think you have to define a serial of choosing them.
Suppose, if you choose in the serial w.k.,a.r.,batsmen and bowler
then, it will be different from the order a.r.,batsmen,bowler and w.k.
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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Fahim Shahriar
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Re: Fantasy Cricket

Unread post by Fahim Shahriar » Sun Sep 16, 2012 4:07 pm

:roll: I should mention it earlier.
I chose this serial. --Batsmen,Bowler,A.R , W.K and at last $12^{th}$ man--
Name: Fahim Shahriar Shakkhor
Notre Dame College

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nafistiham
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Re: Fantasy Cricket

Unread post by nafistiham » Sun Sep 16, 2012 11:37 pm

OK. thanks for that.

solution:
For $4$ batsmen you have $5^4 \cdot _{4}^{12}\textrm{P}$ choices
for $3$ bowlers you have $5^3 \cdot _{3}^{8}\textrm{P}$ choices
for $3$ allrounders you have $4^3 \cdot _{3}^{5}\textrm{P}$ choices
for $1$ wicket keeper you have $1^1 \cdot _{1}^{2}\textrm{P}$ choices
for the $12^{th}$ player you have $12$ choices
multiply them :D
by the way $ _{m}^{n}\textrm{P}=\frac {n!}{(n-m)!}$
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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