Posted by Jane on .
1) In how many ways can 12 distinct objects be separated into three equal piles?
2) In how many ways can 25 students be assigned to 4 distinguishable study groups if at least 6 students must be in each group?
3) in how many ways can 4 students be assigned to 9 class periods if:
a) there are no restrictions?
b) each students must go to a different period?

Data management 
MathMate,
1.
There are n=12! ways to line up 12 distinct object into 12 ordered positions.
If positions 1,2,3 form one pile, then N would be overcounted by 3! times. Therefore n must be divided by 3! for each group of 3.
The total number of ways is therefore
12!/(3!3!3!3!)
2. Follow the same argument as in Q1 to place 24 (distinct) students into 4 groups of 6. There are 4 ways to place the 25th student, so multiply by 4.
3.
No restriction:
We give the choice to the students.
Number of choices for the first student=9
number of choices for the second student=9
.....
number of choices for the fourth student=9
Use the multiplication rule to establish the total number of ways.
3.
With restriction that they go to a different session:
Number of choices for the first student=9
Number of choices for the second student = 8
.....
number of choices for the 4th student = 6
Use the multiplication principle to get the total number of arrangements.