Final answer:
The student's question involves calculating the number of combinations to assign 12 volleyball players into either four triple rooms or six double rooms. This is solved using the combinatorial formula for combinations, considering that rooms are identical and the order in which players are assigned to rooms does not matter.
Step-by-step explanation:
The question asks us to calculate the number of ways 12 volleyball players can be assigned to either four triple rooms or six double rooms. This is a problem of combinatorics, an area of mathematics that deals with counting, permutation, and combination.
a) Assigning 12 players to four triple rooms:
To assign 12 players to four triple rooms, we consider the rooms to be identical, and the order of players in the room doesn't matter. We can calculate this using the formula for combinations:
C(n, k) = n! / (k!(n-k)!)
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where n is the total number of items, and k is the number of items to choose. So for the first room:
C(12, 3) = 12! / (3!(12-3)!) = 220
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For the remaining players and rooms, we continue multiplying the combinations, but we must remember that each time we fill a room, there are fewer players and rooms left.
b) Assigning 12 players to six double rooms:
Similarly, to assign 12 players to six double rooms, we use combinations, but this time we calculate the placement for two players in a room. For the first room:
C(12, 2) = 12! / (2!(12-2)!) = 66
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As with the triple rooms, we continue the process until all players have been assigned.
These calculations show the potential combinations of player assignments to rooms, reflecting the probability and permutations involved in such assignments.