Final answer:
The game with ten students tossing a Frisbee without repeating a partner will end after 45 tosses, determined by the sum of the first nine positive integers.
Step-by-step explanation:
The student's question involves a combinatorial problem that can be approached as a counting exercise in mathematics. When considering the number of tosses needed in the game where ten students are standing in a circle and are throwing a Frisbee to each other without repeating a throw to the same person, we can devise a systematic way to determine the answer.
Starting with one student, they have 9 options for the first toss since they can throw it to any other student except themselves. After the first student tosses the Frisbee, the recipient has 8 options (all the remaining students except for themselves and the one who just tossed the Frisbee). This pattern continues, reducing the number of options by one with each toss until there is only one option left.
To calculate the total number of tosses, we sum the number of options at each step:
9 + 8 + 7 + ... + 1 this is essentially summing the integers from 1 to 9. The sum of the first 'n' positive integers is given by the formula n(n+1)/2. Using this formula, we get: 9(9+1)/2 = 9(10)/2 = 45. Therefore, there will be a total of 45 tosses before the game ends.