1) How many ways can 12 separate objects be separated into three equal piles? 2) How many ways can 25 students be assigned to 4 separate study groups if at least 6 students must be in each group?
3) how many ways can 4 students be assigned to 9 lesson periods if:
a) there are no restrictions?
b) each student must go to a different period? 1.
There are years = 12! ways to align 12 separate objects in 12 ordered positions.
If positions 1,2,3 form a stack, then N would be overcounted by 3! time. So n must be divided by 3! for each group of 3.
The total number of channels is therefore
12!/(3!3!3!3!)
2. Follow the same argument as in Q1 to place 24 (separate) students into 4 groups of 6. There are 4 ways to place the 25th student, so multiply by 4.
3.
No restriction:
We leave the choice up 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 lanes.
3.
With the restriction that they go to a different session:
Number of choices for the first student=9
Number of choices for second student = 8
…..
number of choices for the 4th student = 6
Use the principle of multiplication to get the total number of arrangements.