Administrator

Answer is indeed 27!/[(3!^9)9!]
Let the set of students be {1, 2, 3, 4, ...., 27}
Consider the following division into 9 teams
{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, {13, 14, 15}, {16, 17, 18}, {19, 20, 21}, {22, 23, 24}, {25, 26, 27}}
And compare it with
{{4, 5, 6}, {1, 2, 3}, {7, 8, 9}, {10, 11, 12}, {13, 14, 15}, {16, 17, 18}, {19, 20, 21}, {22, 23, 24}, {25, 26, 27}}
Note that above two are giving us same division of 27 people into 9 teams but they count as two different arrangements in 27!. Likewise there are 9! arrangements, all resulting in the same division. So, the number 27! must be divided by 9!.
Also,
Consider the following division into 9 teams again
{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, {13, 14, 15}, {16, 17, 18}, {19, 20, 21}, {22, 23, 24}, {25, 26, 27}}
And compare it with
{{2, 1, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, {13, 14, 15}, {16, 17, 18}, {19, 20, 21}, {22, 23, 24}, {25, 26, 27}}
Again the above two are two different arrangements in 27! but result in same team formations. Likewise, there are 3!^9 such arrangements that result in the same team formations. So, the number must also be divided by 3!^9.
Thus, the correct answer is 27!/9!*(3!^9)
