document.write( "Question 1023651: If 20 premier league are present and five of the team represent one company the find the number of ways pair of team representing different companies can play a game \n" ); document.write( "

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\n" ); document.write( "Question:
\n" ); document.write( "If 20 premier leagues are present and five of the teams represent one company then find the number of ways a pair of teams representing different companies can play a game.
\n" ); document.write( "
\n" ); document.write( "Solution:
\n" ); document.write( "There are C(20,2) different matches of two teams can be made, out of which C(5,2) are from the same company, where C(n,r)=n!/(r!(n-r)!).
\n" ); document.write( "So C(20,2)-C(5,2)=190-10=180.
\n" ); document.write( "
\n" ); document.write( "When we solve permutations, we need to look for different ways of looking at the same problem. If the results come up the same, this is a verification, or a combinatorial proof that the two expressions are identical.
\n" ); document.write( "
\n" ); document.write( "Here's another aspect of the same problem.
\n" ); document.write( "Assume two cases,
\n" ); document.write( "A. the company has no team in the matches.
\n" ); document.write( "So there are C(15,2)=15*14/2=105 matches.
\n" ); document.write( "B. the company has a team in the matches. Then one particular team of the company can play 15 adversaries, therefore the five teams of the company can form 15*5=75 matches.
\n" ); document.write( "Adding the two cases, we have
\n" ); document.write( "number of ways = 105+75 = 180 as before. \n" ); document.write( "

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