T20 World Cup 2014 is being held in Bangladesh.
Next to Mirpur stadium, there are $24$ flags standing in a row. The flags belong to $9$ different countries.
There are $10$ Australian flag. It's the highest number of flags a country has in that row.
The next highest number of flags that belong to the same country is $4$.
Pakistan and India have the same number of flags, but the number of flags each of them have is less than $4$.
How many permutations are possible for the flags?
(It is possible for a country to have $0$ flags)
Problem Setter: Saumitra da.
Permutation of Flags
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Re: Permutation of Flags
There are \(10\) Australian flag . So , \(14\) flags belong to \(8\) different countries . Since , at least one country has \(4\) flags , there are three cases :-
Case 1 : \(1\) country has \(4\) flags .
So, \(1\) country among \(6\) countries (excluding Pakistan and India )can have these \(4\) flags in \(6\) ways . Then , the other \(10\) flags belong to \(7\) countries (including Pakistan and India ) . Now applying stars and bars , we get the following cases :
Case 1.1 : Pakistan = India = \(0\) . Then, \(5\) countries have \(10\) flags . The number of ways are :
\( \displaystyle \binom{5 + 10 - 1}{10} = 1001\)
Case 1.2 : Pakistan = India = \(1\) . Then, \(5\) countries have \(8\) flags . The number of ways are :
\( \displaystyle \binom{5 + 8 - 1}{8} = 495\)
Case 1.3 : Pakistan = India = \(2\) . Then, \(5\) countries have \(6\) flags . The number of ways are :
\( \displaystyle \binom{5 + 6 - 1}{6} = 210\)
Case 1.4 : Pakistan = India = \(3\) . Then, \(5\) countries have \(4\) flags . The number of ways are :
\( \displaystyle \binom{5 + 4 - 1}{4} = 70\)
The number of permutation for this case is \( 6(1001 + 495 + 210 + 70) = 10656\) .
Case 2 : \(2\) countries have \(4\) flags each .
So, \(2\) countries among \(6\) countries (excluding Pakistan and India )can have these \(8\) flags in \( ^6P_2\) or \(30\) ways . Then , the other \(6\) flags belong to \(6\) countries (including Pakistan and India ) . Now applying stars and bars , we get the following cases :
Case 2.1 : Pakistan = India = \(0\) . Then, \(4\) countries have \(6\) flags . The number of ways are :
\( \displaystyle \binom{4 + 6 - 1}{6} = 84\)
Case 2.2 : Pakistan = India = \(1\) . Then, \(4\) countries have \(4\) flags . The number of ways are :
\( \displaystyle \binom{4 + 4 - 1}{4} = 35\)
Case 2.3 : Pakistan = India = \(2\) . Then, \(4\) countries have \(2\) flags . The number of ways are :
\( \displaystyle \binom{4 + 2 - 1}{2} = 10\)
Case 2.4 : Pakistan = India = \(3\) . Here's just \(1\) permutation in this case .
The number of permutation for this case is \( 30(84 +35 + 10 + 1) = 3900\) .
Case 3 : \(3\) countries have \(4\) flags each .
So , \(3\) countries among \(6\) countries can have these \(12\) flags in \( ^6P_3\) or \(120\) ways . So , the other \(2\) flags belong to \(5\) countries . Again applying stars and bars , we get the following cases :
Case 3.1 : Pakistan = India = \(0\) . Then, \(3\) countries have \(2\) flags . The number of ways are :
\( \displaystyle \binom{3 + 2 - 1}{2} = 6\)
Case 3.2 : Pakistan = India = \(1\) . Here's just \(1\) permutation in this case .
The number of permutation for this case would be \( 120(6 + 1) = 840\) .
Hence , the total number of possible permutation is \( 10656 + 3900 + 840 = 15396\) .
Case 1 : \(1\) country has \(4\) flags .
So, \(1\) country among \(6\) countries (excluding Pakistan and India )can have these \(4\) flags in \(6\) ways . Then , the other \(10\) flags belong to \(7\) countries (including Pakistan and India ) . Now applying stars and bars , we get the following cases :
Case 1.1 : Pakistan = India = \(0\) . Then, \(5\) countries have \(10\) flags . The number of ways are :
\( \displaystyle \binom{5 + 10 - 1}{10} = 1001\)
Case 1.2 : Pakistan = India = \(1\) . Then, \(5\) countries have \(8\) flags . The number of ways are :
\( \displaystyle \binom{5 + 8 - 1}{8} = 495\)
Case 1.3 : Pakistan = India = \(2\) . Then, \(5\) countries have \(6\) flags . The number of ways are :
\( \displaystyle \binom{5 + 6 - 1}{6} = 210\)
Case 1.4 : Pakistan = India = \(3\) . Then, \(5\) countries have \(4\) flags . The number of ways are :
\( \displaystyle \binom{5 + 4 - 1}{4} = 70\)
The number of permutation for this case is \( 6(1001 + 495 + 210 + 70) = 10656\) .
Case 2 : \(2\) countries have \(4\) flags each .
So, \(2\) countries among \(6\) countries (excluding Pakistan and India )can have these \(8\) flags in \( ^6P_2\) or \(30\) ways . Then , the other \(6\) flags belong to \(6\) countries (including Pakistan and India ) . Now applying stars and bars , we get the following cases :
Case 2.1 : Pakistan = India = \(0\) . Then, \(4\) countries have \(6\) flags . The number of ways are :
\( \displaystyle \binom{4 + 6 - 1}{6} = 84\)
Case 2.2 : Pakistan = India = \(1\) . Then, \(4\) countries have \(4\) flags . The number of ways are :
\( \displaystyle \binom{4 + 4 - 1}{4} = 35\)
Case 2.3 : Pakistan = India = \(2\) . Then, \(4\) countries have \(2\) flags . The number of ways are :
\( \displaystyle \binom{4 + 2 - 1}{2} = 10\)
Case 2.4 : Pakistan = India = \(3\) . Here's just \(1\) permutation in this case .
The number of permutation for this case is \( 30(84 +35 + 10 + 1) = 3900\) .
Case 3 : \(3\) countries have \(4\) flags each .
So , \(3\) countries among \(6\) countries can have these \(12\) flags in \( ^6P_3\) or \(120\) ways . So , the other \(2\) flags belong to \(5\) countries . Again applying stars and bars , we get the following cases :
Case 3.1 : Pakistan = India = \(0\) . Then, \(3\) countries have \(2\) flags . The number of ways are :
\( \displaystyle \binom{3 + 2 - 1}{2} = 6\)
Case 3.2 : Pakistan = India = \(1\) . Here's just \(1\) permutation in this case .
The number of permutation for this case would be \( 120(6 + 1) = 840\) .
Hence , the total number of possible permutation is \( 10656 + 3900 + 840 = 15396\) .
Last edited by sadman sakib on Sat Mar 29, 2014 7:12 pm, edited 1 time in total.
Re: Permutation of Flags
I think you got the problem wrong.
Australia has 10 flags. Another country has 4 flags. This country is neither India, nor Pakistan. It is the second highest.
The last 10 flags belong to the other 7 teams (which include India and Pakistan).
We are looking for the number of permutations possible.
I think the ans is a big number. You can change the number of second highest from 4 to 7.
Australia has 10 flags. Another country has 4 flags. This country is neither India, nor Pakistan. It is the second highest.
The last 10 flags belong to the other 7 teams (which include India and Pakistan).
We are looking for the number of permutations possible.
I think the ans is a big number. You can change the number of second highest from 4 to 7.
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
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"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
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Re: Permutation of Flags
The line of confusion is :
From your post it seems that you wanted to say : The next highest number of flags that belong to at least \(1\) country is \(4\) .Those two statements are quite different .
I just edited the answer accordingly . I think this is the right answer .
'same' always means more than \(1\) ,doesn't it ?Labib wrote:The next highest number of flags that belong to the same country is \(4\)
From your post it seems that you wanted to say : The next highest number of flags that belong to at least \(1\) country is \(4\) .Those two statements are quite different .
I just edited the answer accordingly . I think this is the right answer .
Re: Permutation of Flags
Only one country has 4 flags.
I'll ask Saumitra da to check the solution.
I'll ask Saumitra da to check the solution.
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
Re: Permutation of Flags
Saumitra da says, more than one country can have 4 flags.Labib wrote:Only one country has 4 flags.
I'll ask Saumitra da to check the solution.
Saumitra da says, even according the given statement, the logics you used in some cases were wrong.sadman sakib wrote:
Try harder
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes