Final answer:
The gene responsible for the genetic disorder in the family is most likely recessive because out of six homozygous recessive individuals, only three have the disease, which means others might be carriers.
Step-by-step explanation:
If six people in a family of ten are homozygous for a recessive allele which causes a genetic disorder, but only three have the disease, this means the gene responsible for the disease is most likely recessive. With autosomal recessive inheritance, a person must inherit two copies of the recessive allele (one from each parent) to exhibit the disorder. If someone is homozygous recessive, they will display the disorder's phenotype.
However, if they are heterozygous, with one dominant and one recessive allele, they will not show the disorder and will be considered carriers. Since six are homozygous and there are more individuals who are homozygous recessive than those expressing the disorder, it's possible that some family members could be carriers without expressing the disease phenotype.