Question

I was practicing my problem-solving and was a bit confused as to why I supposedly got a wrong answer. Below is what I was solving:

Two families are planning to go on a canoe trip together. The families consist of the following people: Robert and Mary Henderson and their three sons Tommy, Dan, and William; Jerome and Ellen Penick and their two daughters Kate and Susan. There will be three canoes, with three people in each canoe. At least one of the four parents must be in each canoe. At least one person from each family must be in each canoe. Question: If each of the Henderson children ride in a different canoe, which of the following must be true? (I) The Penick children do not ride together (II) The Penick parents do not ride together (III) The Henderson parents do not ride together

a) I only
b) II only
c) I and II only
d) I and III only

My answer is d) but apparently the correct answer might be a).

So let's assume that III is false (that the Henderson parents do ride together).
Then the statements "at least one person from each family must be in each canoe" and "each of the Henderson children ride in a different canoe" cannot both be followed since there are 3 canoes and 3 Henderson children. But if a Henderson child rides in the same canoe containing both Henderson parents, then it is full and does not contain a member of the other family.

Can anyone confirm if I'm right or wrong? Thanks.