Puzzle -Daughter's Age Problem

A census taker came to a house where a man lived with
three daughters. "What are your daughters' ages?" he asked.
The man replied, "The product of their ages is 72, and the
sum of their ages is my house number."
"But that's not enough information," the census taker insisted.
"All right," answered the farmer, "the oldest loves chocolate.

What are the daughters' ages?

Solution :
We can look at all the possibilities this way - the oldest may be the same age as the middle, and the middle may be the same age as the youngest:
OLD MID YOUNG SUM
72 1 1 74
36 2 1 39
24 3 1 28
18 4 1 23
18 2 2 22
12 3 2 17
12 6 1 19
9 8 1 18
9 4 2 15
8 3 3 14
6 6 2 14
6 4 3 13
These are all the possibilities, since if we give the oldest a lower age, there is no way the product can equal 72.We know their ages add up to the farmer's house number. The sum column gives all possibilities for the house number. When the farmer gave the census person the information about the product of their ages being 72 and the sum of their ages being his house number the census person said this was not enough information. So there must have been at least two different possibilities that were still viable options for the census taker to choose from. This means the house number must have had two sums equal it. So the daughters are:
8 3 3 14
6 6 2 14
since they both add up to 14, the only number that appeared twice in the sum of the ages. The fact that the farmer had an oldest daughter says that the daughters must in fact be ages 8, 3, and 3.