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There are 9 students in Ms. Q's class. Ms. Q wants to split up her class into three groups for a group project, and wants each group to have at least two students. In how many ways can she do this?

Jul 13, 2023

#1
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There are three cases to consider:

Case 1: Each group has 2 students. Then there are (29​)=36 ways to form the groups.

Case 2: One of the groups has 3 students and the other two groups have 2 students each. Then there are (39​)=84 ways to choose which student is in the group of 3, and then 3 ways to order the 3 students in that group.

Case 3: One of the groups has 4 students and the other two groups have 1 student each. Then there are (49​)=126 ways to choose which student is in the group of 4, and then 4 ways to order the 4 students in that group.

Summing the number of ways in each case, we get a total of 36+84+126=246​ ways to form the groups.

Jul 13, 2023
#2
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There are 9 students in total. In your answer, in case 1 there are 6 students, in case 2 7 students, and in case 3 there are 6 students.

Guest Jul 13, 2023
#3
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1 - (9 C 2) * (7 C 2) / 2!   =378
2 - (9 C 2) * (7 C 3)         =1260
3 - (9 C 3) * (6 C 3) / 3!   =280

Total number of way=378 + 1260 +280=1918 ways.

Jul 15, 2023