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  • A sports tournament has 10 teams, and each team will play against every other team exactly once. In how many different ways can the schedule be arranged?  

A sports tournament has 10 teams, and each team will play against every other team exactly once. In how many different ways can the schedule be arranged?

 

Option: 1

35 ! / 35


Option: 2

45 ! / 45


Option: 3

55 ! / 55


Option: 4

25 ! / 25


Answers (1)

best_answer

To calculate the number of different ways the schedule can be arranged for a sports tournament with 10 teams, where each team plays against every other team exactly once, we can use the concept of permutations.

Since each team will play against every other team exactly once, we can think of the schedule as a sequence of matches, where each match involves two distinct teams.

The total number of matches in the tournament can be calculated using the combination formula C(10,2)=45, which represents the number of ways to select 2 teams out of 10 teams.

Now, we need to consider the order in which these matches can be scheduled. Since each match involves two teams, we have 45 matches to schedule.

The number of different ways to arrange the schedule for these 45 matches can be calculated using the concept of permutations. The number of permutations of 45 matches can be calculated as 45 !. Therefore, the total number of different ways the schedule can be arranged is:
45 !.

However, the ordering of the matches within each round doesn't matter since each team will play against every other team exactly once. Therefore, we need to account for the rotational symmetry in the schedule.

Since there are 45 matches in the schedule, the rotational symmetry factor is 45 , which means each schedule can be rotated to create an equivalent schedule.
To account for this rotational symmetry, we divide the total number of arrangements by 45 .

Therefore, the final number of different ways the schedule can be arranged for the sports tournament with 10 teams is:

45 ! / 45=45 \times 44 \times 43 \times \ldots \times 3 \times 2 \times 1 .
Calculating this value precisely would require a very large number of calculations, but it is an extremely large number.

Therefore, there are an extremely large number of different ways the schedule can be arranged for the sports tournament with 10 teams.

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Gaurav

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