$10\sqrt{10}=(\sqrt{100})(\sqrt{10})=\sqrt{1000}$

1000 = (2)(2)(2)(5)(5)(5)

Let's use the above prime factorization to see see how many different perfect squares we can factor out 1000.

For example, 1000 = 4 x 250 (where 4 is a perfect square)
So, $\sqrt{1000} = \sqrt{4} \times \sqrt{250} = 2\sqrt{250}$
In this case, $a = 2$ and $b = 250$, which means $a + b = 2 + 250 = 252$
So, answer choice B works.

Also, 1000 = 25 x 40 (where 25 is a perfect square)
So, $\sqrt{1000} = \sqrt{25} \times \sqrt{40} = 5\sqrt{40}$
In this case, $a = 5$ and $b = 40$, which means $a + b = 5 + 40 = 45$
So, answer choice A works.

What about answer choice C?
The only way to get CLOSE to a sum of 1000, is to factor out a 1 from 1000
In other words, let's examine 1000 = 1 x 1000 (where 1 is a perfect square)
Here, $\sqrt{1000} = \sqrt{1} \times \sqrt{1000} = 1\sqrt{1000}$
In this case, $a = 1$ and $b = 1000$, which means $a + b = 1 + 1000 = 1001$
Close, but no good.

Answer: A, B

Cheers,
Brent