Question

design a circuit that counts only 1 invalid input 1111,output is 00, valid input 1110 and output is 10 valid input 1010output is 11

Answer 1

Is there something wrong in the question ?  
    valid input  1110,   output should be   11
                       1010                                 10.

see the diagram for truth table,  Karnaugh map, then the Logic functions in terms of variables.   Then the combinatorial circuit with gates  AND, OR, NOT.

If exclusive OR gates could be used , then circuit will be simpler.

X = ABC' + AC'D + BC'D + A'CD + BCD' + AB'C 
    = A(B'C + C'B) + B (C'D + D'C) + D (A C' + A' C) 

[ALSO, X' = A'B'C' + A'B' D' + A'C'D' + B'C'D' + ABCD
       By  taking the zeros in the table.]

Y = (A'B + A B') C'D' + (A'B' + AB) C'D + CD (A'B + A B')+ (A'B'+AB)CD' 
   = (A'B + A B') (C'D' + CD) + ( A'B' + AB) (C'D + C D')