Regular Expressions for Strings with Specific Character Patterns

[kliker]

Homework Statement

Σ = {a,b}
find regular expression for

1)all strings that haven't got more than 3 a's
2)all strings that have a number of a's that is divisible by 3
3)all strings that have the substring aaa only one time

The Attempt at a Solution

1) b*Ub*ab*Ub*ab*ab*Ub*ab*ab*ab*

2) (b*Ub*ab*ab*ab*)*

3) b*(aUaaUe)b*aaab*(aUaaUe)b*

can you please tell me if I am correct? this is not homework actually but i didnt know where to post it, I am just preparing myself for the upcoming exams and need to know if i have these correct

thanks in advance

 

[kliker]

i have one more

Σ = {a,b,c}

i want a regular expression for all strings that contain at least 1 a, 1 b and 1 c

ok my solution is this

(a(bUc)*Ub(aUc)*Uc(aUb)*)*

but the book has this solution

(aUbUc)*c(aUbUc)*b(aUbUc)*a(aUbUc)*

which is right but is my solution wrong?