Give a regular expression of all strings containing no more

Give a regular expression of all strings containing no more than three a\'s for the language on the alphabet, ={a, b, c}.

Solution

(a) all strings containing exactly one a.
Solution
(b + c)*a(b + c)*
(b) all strings containing no more than three a\'s.
Solution
(b + c)*(a + lambda)(b + c)*(a + lambda))(b + c)*(a + lambda)(b + c)*
(d) all strings that contain no run of a\'s of length greater than two.
Solution
(b + c)*+ (b + c)*((a + aa)(b + c)+)*(a + aa)(b + c)*
(e) all strings in which all runs of a\'s have lengths that are multiples
of three.
Solution
(b + c)*((aaa)(b + c)*)*