Brain Teaser: Find the ages

Question:
I was visiting a friend one evening and remembered that he had three daughters. I asked him how old they were. "The product of their ages is 72," he answered. Quizzically, I asked, "Is there anything else you can tell me?" "Yes," he replied, "the sum of their ages is equal to the number of my house." I stepped outside to see what the house number was. Upon returning inside, I said to my host, "I'm sorry, but I still can't figure out their ages." He responded apologetically, "I'm sorry, I forgot to mention that my oldest daughter likes strawberry shortcake." With this information, I was able to determine all of their ages. How old is each daughter?

Answer:
The first clue is that the product of their ages is 72. Therefore, the list of all possible sets of numbers whose product is 72 is:

72 1 1
36 2 1
24 3 1
18 4 1
18 2 2
12 3 2
12 6 1
9 8 1
9 4 2
8 3 3
6 6 2
6 4 3

The second clue involves the sums of their ages which are:

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

Out of all these sums, two of them are equal:

8 + 3 + 3 = 14
6 + 6 + 2 = 14

The final clue is that his "oldest" daughter likes cake, which means that we can eliminate the second one of the above. Therefore, his daughter's ages are:

8 + 3 + 3 = 14

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