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Community mathematics theme
1) five clubs take 1:C(5 1) and two students, so the combination is c (51) * c (3,2).

Another 1 student joined one of the remaining four clubs, C(4 1).

Each student has 5 minutes to choose, and 3 students choose: 5 3.

So the probability is: c (5,1) * c (3,2) * c (4, 1)/5 3 = 12/25.

2)

No students join the A or B clubs.

C (4 4,3 3) * p (3 3,3)/53 =24/ 125, probability: = 24/125.

There are 1 student, and 2 students participate in others: C (3, 1) * C (2,1) * p (4,2),

Probability: c (3, 1) * c (2,1) * p (4,2)/53 = 72/125.

Two students participated, 1 student participated in others: c (3,2) * c (2, 1) * C (4, 1).

Probability: c (3,2) * c (2, 1) * p (4, 1)/5 3 = 6/ 125.

There are three students, C (3,3) * C (2, 1).

Probability: c (3,3) * c (2, 1)/5 3 = 2/ 125.

Mathematical expectation, I think you can work it out.