Probably ellos saben las soluciones, but context matters!
"They" is the plural, 3rd person subject → ellos or ellas.
"Know" comes from to know. In this case, they know answers, which indicates that they know a fact (saber) instead of being familiar with something/someone (conocer). I'd write saben here.
"The answers" can mean a few things. Are these answers to questions (las respuestas)? Solutions to a problem (las soluciones)? I'd recommend rereading the text around this sentence to determine the intended meaning more precisely.