Final answer:
A DNA test can determine if the sister is a carrier of Fragile X syndrome, which is an X-linked recessive disorder. The father cannot be a carrier, but he can pass the disease to his daughters if he is affected. The brother inherited Fragile X from their mother, not the sister.
Step-by-step explanation:
Fragile X syndrome is an X-linked recessive disorder, which means that males with one affected X chromosome (hemizygous) will express the disease while females need to have two affected chromosomes (homozygous) to be affected. Since females have two X chromosomes, they can either be unaffected carriers if they have one affected and one unaffected X chromosome, or they may be affected if both X chromosomes carry the mutation.
A DNA test can determine whether the student's sister is a carrier for Fragile X. If the sister has one mutated and one normal X chromosome, she is a carrier. If both are mutated or both are normal, she would either be affected by or unaffected by the disease, respectively. The father cannot be a carrier because males have one X and one Y chromosome and any mutation on the X chromosome results in the disease.
The sister cannot influence whether her brother inherits Fragile X, as her brother would have received his X chromosome from their mother. So, the brother inherited Fragile X from the mother, not from the sister. The father's role in the inheritance of X-linked disorders relates to his daughters; if he has the disease, all his daughters would be carriers since they would inherit his only X chromosome.
Therefore, based on the genetic testing results, we could determine if the sister is a carrier of Fragile X (a), but not if she will develop the disease (c), since women can be carriers without being affected. The father's status as a carrier is not applicable for X-linked disorders (b), and the brother cannot inherit the disease from his sister, but rather the mother (d).