Tree diagrams are really helpful in solving such probability questions.
Let Y represent a person having the illness and let N represent a person not having the illness.
There can be three possibilities of exactly one person having such an illness.
YNN, NYN, NNY
25% of the people of Wales have reported such an illness.
We only need to find he probability of one of the possibilities first.
Let first person have the illness and the next two persons not have the illness.
Probability of first person having the illness = ¼
Probability of second person not having the illness = ¾
Probability of third person not having the illness = ¾
¼ x ¾ x ¾ = ⁹⁄₆₄
Since there are 3 possibilities, we need to multiply this probability by 3.
⁹⁄₆₄ x 3 = ²⁷⁄₆₄