For a final project of my graduate course I have to split my 20 students into 5

Question

For a final project of my graduate course I have to split my 20 students into 5 teams of 4 a) How many ways do I have to form the teams? b) Additionally, since each project requires a student to be the modeler, one student to be the reporter, one student to solve the models, and one student to do outreach activities. How many total ways could these teams be formed, assuming that students have already been assigned to the teams? c What would change in your analyses, if you consider the problem as a sequence of steps? (Note, assign students, and then assign the roles)

Explanation / Answer

Answer:

a
We have to form the team primary and after that allocate roles
ways of form a team of 4 = 20C4x16C4x12C4x8C4x4C4 {selecting 4 students out of 20 then select 4 out of residual 16 and so on}
ways of transmission the role for each side = 4C1x3C1x2C1x1C1 {1 student to be assigned out of 4 roles and then 1 student to be assign out of 3 roles and so on}
Ways of transmission the roles for 5 teams = 5x (4C1x3C1x2C1x1C1)
Hence total ways = 20C4x4C1x3C1x2C1x1C1x16C4x4C1x3C1x2C1x1C1x12C4x4C1x3C1x2C1x1C1x8C4x4C1x3C1x2C1x1C1x4C4x4C1x3C1x2C1x1C1
Or,
20C4x16C4x12C4x8C4x4C4x5x (4C1x3C1x2C1x1C1)

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