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m = DeleteDuplicates[Map[Sort, Tuples[{a, b, c, d}, 3]]];

DeleteCases[m, {Alternatives@@Map[OrderlessPatternSequence[_, #, #] &, {b, c, d}]}]

(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d},{b, c, d}}*)

(* adopting Wizard's approach *)
DeleteCases[m, {OrderlessPatternSequence[x : b | c | d, x_, _] }] 
(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d}, {b, c, d}}*)

m = DeleteDuplicates[Map[Sort, Tuples[{a, b, c, d}, 3]]];

DeleteCases[m, {Alternatives@@Function[x, OrderlessPatternSequence[_, x, x],
Listable], {b, c, d}]}]

(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d},{b, c, d}}*)

(* adopting Wizard's approach *)
DeleteCases[m, {OrderlessPatternSequence[x : b | c | d, x_, _] }] 
(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d}, {b, c, d}}*)

m = DeleteDuplicates[Map[Sort, Tuples[{a, b, c, d}, 3]]];

DeleteCases[m, {OrderlessPatternSequence[_, b, b] | 
OrderlessPatternSequence[_, c, c] | OrderlessPatternSequence[_, d, d]}]

(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d},{b, c, d}}*)

(* adopting Wizard's approach *)
DeleteCases[m, {OrderlessPatternSequence[x : b | c | d, x_, _] }] 
(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d}, {b, c, d}}*)

m = DeleteDuplicates[Map[Sort, Tuples[{a, b, c, d}, 3]]];

DeleteCases[m, {Alternatives@@Map[OrderlessPatternSequence[_, #, #] &, {b, c, d}]}]

(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d},{b, c, d}}*)

(* adopting Wizard's approach *)
DeleteCases[m, {OrderlessPatternSequence[x : b | c | d, x_, _] }] 
(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d}, {b, c, d}}*)

DeleteCases[m, {OrderlessPatternSequence[_, b, b] | 
OrderlessPatternSequence[_, c, c] | OrderlessPatternSequence[_, d, d]}]

(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d},{b, c, d}}*)

(* adopting Wizard's approach *)
DeleteCases[m, {OrderlessPatternSequence[x : b | c | d, x_, _] }] 
(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d}, {b, c, d}}*)

m = DeleteDuplicates[Map[Sort, Tuples[{a, b, c, d}, 3]]];

DeleteCases[m, {OrderlessPatternSequence[_, b, b] | 
OrderlessPatternSequence[_, c, c] | OrderlessPatternSequence[_, d, d]}]

(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d},{b, c, d}}*)

(* adopting Wizard's approach *)
DeleteCases[m, {OrderlessPatternSequence[x : b | c | d, x_, _] }] 
(*{{a, a, a}, {a, a, b}, {a, a, c}, {a, a, d}, {a, b, c}, {a, b, d}, {a,c,d}, {b, c, d}}*)