Different values of $x$ and $y$ between $\sqrt{39}$ and $\sqrt{224}$

Question

If $x$ and $y$ are whole numbers between $\sqrt{39}$ and $\sqrt{224}$, then how many different values can $x$ + $y$ have?

OK, first I found that the set numbers are: $$7, 8 ,9 ,10 ,11 ,12, 13,14$$ Since they are all between $\sqrt{39}$ and $\sqrt{224}$. So then I said $7 +8$, $7+9$ all the way up to $14+14$ and I found that there are 28 different values but the answer sheet says the correct answer is $15$. How is this? Note that I did not include $8 + 7$, I said it's the same as $7+8$. E.g: After I got to $7+14$ I started the next number with $8+8$ and then $9+9$ etc. Incase anyone will suggest that I counted the same number twice.

Answer 1

The smallest sum is 14. The largest is 28. You also get every number in between. There are 15 such numbers.

Answer 2

You've correctly obtained $x\in[7,14]$. Thus, the max. sum is $28$, and the min. is $14$. When adding up, you get all whole number values between $14$ and $28$, i.e., $15$ numbers.