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Question Video: Applications of the Counting Principle (Product Rule with Replacement) Mathematics • 12th Grade

In a football league, a team plays every other team twice. If there are 14 teams in the league, determine the total number of matches to be played.

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Video Transcript

In a football league, a team plays every other team twice. If there are 14 teams in the league, determine the total number of matches to be played.

One way of answering this question is using permutations. In this question, we’ll be dealing with permutations without repetition where order matters. If we let the 14 teams be represented by the letters A to N, and we know that each team is playing every other team twice, then we need to calculate 14P two. Recalling that 𝑛P𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial, then 14P two is equal to 14 factorial divided by 12 factorial. We can simplify the numerator to 14 multiplied by 13 multiplied by 12 factorial. Dividing the numerator and denominator by 12 factorial gives us 14 multiplied by 13. This is equal to 182. If there are 14 teams in a football league and each team plays every other team twice, there will be a total of 182 matches.

An alternative method here would be to use the fundamental counting principle. As each team is playing every other team twice, they will play home and away. For example, team A will be home to team B and also during the season, team B will be home to team A. This means that for every match, there are 14 possible home teams. Once that home team has been selected, there are 13 possible away teams. We can then multiply 14 by 13 to calculate the total number of matches. Once again, this gives us an answer of 182.

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