This notebook checks cases with exactly one 4 in some row, 
no 3 in that row, 
    and at most one 4 in elsewhere 
(i . e .  
              in other two rows) .   Cases covered are

422 222 200 000
422 222 110 000
422 221 111 000
422 211 111 100
422 211 111 110
421 111 111 111

a  =  {a1, a2, a3, a4, a5, a6, a7, a8, a9,  a10, a11, 
        a12}

b  =  {b1, b2, b3, b4, b5, b6, b7, b8, b9,  b10, b11, 
        b12}

c  =  {c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12}

L[X_ ,  i_ ,  1]  :=  
      D[X,  a[[i]],  a[[i]],  a[[i]],  a[[i]]];  
    L[X_ ,  i_ ,  2]  :=  
      D[X,  a[[i]],  a[[i]],  a[[i]],  b[[i]]]; 
    L[X_ ,  i_ ,  3]  :=  
      D[X,  a[[i]],  a[[i]],  a[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  4]  :=  
      D[X,  a[[i]],  a[[i]],  b[[i]],  b[[i]]]; 
    L[X_ ,  i_ ,  5]  :=  
      D[X,  a[[i]],  a[[i]],  b[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  6]  :=  
      D[X,  a[[i]],  a[[i]],  c[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  7]  :=  
      D[X,  a[[i]],  b[[i]],  b[[i]],  b[[i]]]; 
    L[X_ ,  i_ ,  8]  :=  
      D[X,  a[[i]],  b[[i]],  b[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  9]  :=  
      D[X,  a[[i]],  b[[i]],  c[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  10]  :=  
      D[X,  a[[i]],  c[[i]],  c[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  14]  :=  
      D[X,  b[[i]],  b[[i]],  b[[i]],  b[[i]]]; 
    L[X_ ,  i_ ,  12]  :=  
      D[X,  b[[i]],  b[[i]],  b[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  11]  :=  
      D[X,  b[[i]],  b[[i]],  c[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  13]  :=  
      D[X,  b[[i]],  c[[i]],  c[[i]],  c[[i]]]; 
    L[X_ ,  i_ ,  15]  :=  
      D[X,  c[[i]],  c[[i]],  c[[i]],  
        c[[i]]];

M = 0;   p1  = 1;  L1  =  L[T, 1, 1];

    For[p2  = 4,  p2  ≤  6,  p2++, 
      L2  =  L[L1,  2,  p2]; 
      Print[p2]; 
For[p3  = p2,  
        p3  ≤  6,  p3++, L3  =  L[L2,  3, p3];  
        Print[M]; 
For[p4  = p3,  
          p4 ≤  6,  p4++, 
          L4  =  L[L3,  4, p4];  
For[p5  = p4,  
            p5 ≤  6,  p5++, 
            L5  = L[L4, 5, p5]; 
For[p6  = p5,  
              p6  ≤  6, p6++, 
              L6  = L[L5, 6, p6]; 
For[p7 = p6,  
                p7 ≤  6,  p7++, 
                L7  = L[L6, 7, p7]; 
For[p8  = 11,  
                  p8  ≤  13,  p8++, 
                  L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                    p9  ≤  13,  p9++, 
                    L9  = L[L8, 9, p9];  
                    For[p10 = p9,  p10  ≤  13,  p10++, 
                      L10  =  L[L9,  10, p10 ];  
For[
                        p11  = p10,  
                        p11  ≤  13,  p11++,  
                        L11  =  L[L10, 11, p11]; 
                        For[p12  = p11,  
                          p12 ≤  14,  p12++,  
                          L12 = L[L11, 12, p12]; 
If[
                            L12 < 0,  
                            Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                              p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                              p12]];  
                          If[L12 > 0,  
                            M  =  
                              M + 1]; 
]]]]]]]]]]];  
    Print[M]; Print[FINISHED];  M = 0;

    For[p2  = 4,  p2  ≤  6,  p2++, 
      L2  =  L[L1,  2,  p2]; 
      Print[p2]; 
For[p3  = p2,  
        p3  ≤  6,  p3++, L3  =  L[L2,  3, p3];  
        Print[M]; 
For[p4  = p3,  
          p4 ≤  6,  p4++, 
          L4  =  L[L3,  4, p4];  
For[p5  = p4,  
            p5 ≤  6,  p5++, 
            L5  = L[L4, 5, p5]; 
For[p6  = p5,  
              p6  ≤  6, p6++, 
              L6  = L[L5, 6, p6]; 
For[p7 = 11,  
                p7 ≤  13,  p7++, 
                L7  = L[L6, 7, p7]; 
For[p8  = p7,  
                  p8  ≤  13,  p8++, 
                  L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                    p9  ≤  13,  p9++, 
                    L9  = L[L8, 9, p9];  
                    For[p10 = p9,  p10  ≤  14,  p10++, 
                      L10  =  L[L9,  10, p10 ];  
For[
                        p11  = 7,  p11  ≤  10,  p11++,  
                        L11  =  L[L10, 11, p11]; 
                        For[p12  = p11,  
                          p12 ≤  10,  p12++,  
                          L12 = L[L11, 12, p12]; 
If[
                            L12 < 0,  
                            Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                              p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                              p12]];  
                          If[L12 > 0,  
                            M  =  
                              M + 1]; 
]]]]]]]]]]];  
    Print[M]; Print[FINISHED];  M = 0;

    For[p2  = 4,  p2  ≤  6,  p2++, 
      L2  =  L[L1,  2,  p2]; 
      Print[p2]; 
For[p3  = p2,  
        p3  ≤  6,  p3++, L3  =  L[L2,  3, p3];  
        Print[M]; 
For[p4  = p3,  
          p4 ≤  6,  p4++, 
          L4  =  L[L3,  4, p4];  
For[p5  = p4,  
            p5 ≤  6,  p5++, 
            L5  = L[L4, 5, p5]; 
For[p6  = 11,  
              p6  ≤  13, p6++, 
              L6  = L[L5, 6, p6]; 
For[p7 = p6,  
                p7 ≤  13,  p7++, 
                L7  = L[L6, 7, p7]; 
For[p8  = p7,  
                  p8  ≤  14,  p8++, 
                  L8  =  L[L7, 8, p8]; 
For[p9  = 7,  
                    p9  ≤  10,  p9++, 
                    L9  = L[L8, 9, p9];  
                    For[p10 = p9,  p10  ≤  10,  p10++, 
                      L10  =  L[L9,  10, p10 ];  
For[
                        p11  = p10,  
                        p11  ≤  10,  p11++,  
                        L11  =  L[L10, 11, p11]; 
                        For[p12  = p11,  
                          p12 ≤  10,  p12++,  
                          L12 = L[L11, 12, p12]; 
If[
                            L12 < 0,  
                            Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                              p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                              p12]];  
                          If[L12 > 0,  
                            M  =  
                              M + 1]; 
]]]]]]]]]]];  
    Print[M]; Print[FINISHED];  M = 0;

    For[p2  = 4,  p2  ≤  6,  p2++, 
      L2  =  L[L1,  2,  p2]; 
      Print[p2]; 
For[p3  = p2,  
        p3  ≤  6,  p3++, L3  =  L[L2,  3, p3];  
        Print[M]; 
For[p4  = p3,  
          p4 ≤  6,  p4++, 
          L4  =  L[L3,  4, p4];  
For[p5  = 11,  
            p5 ≤  13,  p5++, 
            L5  = L[L4, 5, p5]; 
For[p6  = p5,  
              p6  ≤  14, p6++, 
              L6  = L[L5, 6, p6]; 
For[p7 = 7,  
                p7 ≤  10,  p7++, 
                L7  = L[L6, 7, p7]; 
For[p8  = p7,  
                  p8  ≤  10,  p8++, 
                  L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                    p9  ≤  10,  p9++, 
                    L9  = L[L8, 9, p9];  
                    For[p10 = p9,  p10  ≤  10,  p10++, 
                      L10  =  L[L9,  10, p10 ];  
For[
                        p11  = p10,  
                        p11  ≤  10,  p11++,  
                        L11  =  L[L10, 11, p11]; 
                        For[p12  = p11,  
                          p12 ≤  10,  p12++,  
                          L12 = L[L11, 12, p12]; 
If[
                            L12 < 0,  
                            Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                              p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                              p12]];  
                          If[L12 > 0,  
                            M  =  
                              M + 1]; 
]]]]]]]]]]];  
    Print[M]; Print[FINISHED];  M = 0;

    For[p2  = 4,  p2  ≤  6,  p2++, 
      L2  =  L[L1,  2,  p2]; 
      Print[p2]; 
For[p3  = p2,  
        p3  ≤  6,  p3++, L3  =  L[L2,  3, p3];  
        Print[M]; 
For[p4  = 11,  
          p4 ≤  14,  p4++, 
          L4  =  L[L3,  4, p4];  
For[p5  = 7,  
            p5 ≤  10,  p5++, 
            L5  = L[L4, 5, p5]; 
For[p6  = p5,  
              p6  ≤  10, p6++, 
              L6  = L[L5, 6, p6]; 
For[p7 = p6,  
                p7 ≤  10,  p7++, 
                L7  = L[L6, 7, p7]; 
For[p8  = p7,  
                  p8  ≤  10,  p8++, 
                  L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                    p9  ≤  10,  p9++, 
                    L9  = L[L8, 9, p9];  
                    For[p10 = p9,  p10  ≤  10,  p10++, 
                      L10  =  L[L9,  10, p10 ];  
For[
                        p11  = p10,  
                        p11  ≤  10,  p11++,  
                        L11  =  L[L10, 11, p11]; 
                        For[p12  = p11,  
                          p12 ≤  10,  p12++,  
                          L12 = L[L11, 12, p12]; 
If[
                            L12 < 0,  
                            Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                              p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                              p12]];  
                          If[L12 > 0,  
                            M  =  
                              M + 1]; 
]]]]]]]]]]];  
    Print[M]; Print[FINISHED];  M = 0;

    For[p2  = 4,  p2  ≤  5,  p2++, 
        L2  =  L[L1,  2,  p2]; 
        Print[p2]; 
For[p3  = 7,  
          p3  ≤  10,  p3++, L3  =  L[L2,  3, p3];  
          Print[M]; 
For[p4  = p3,  
            p4 ≤  10,  p4++, 
            L4  =  L[L3,  4, p4];  
For[p5  = p4,  
              p5 ≤  10,  p5++, 
              L5  = L[L4, 5, p5]; 
For[p6  = p5,  
                p6  ≤  10, p6++, 
                L6  = L[L5, 6, p6]; 
For[p7 = p6,  
                  p7 ≤  10,  p7++, 
                  L7  = L[L6, 7, p7]; 
For[p8  = p7,  
                    p8  ≤  10,  p8++, 
                    L8  =  L[L7, 8, p8]; 
For[
                      p9  = p8,  p9  ≤  10,  p9++, 
                      L9  = L[L8, 9, p9];  
                      For[p10 = p9,  p10  ≤  10,  p10++, 
                        L10  =  
                          L[L9,  10, p10 ];  
For[
                          p11  = p10,  
                          p11  ≤  10,  p11++,  
                          L11  =  L[L10, 11, p11]; 
                          For[p12  = p11,  
                            p12 ≤  10,  p12++,  
                            L12 = L[L11, 12, p12]; 
If[
                              L12 < 0,  
                              Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                                p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                                p12]];  
                            If[L12 > 0,  
                              M  =  
                                M + 1]; 
]]]]]]]]]]] ;  Print[M] ;  Print[
        FINISHED] ;