If the sum of the digits is 34, then there are only 2 possible cases:
Case i: The 4 digits are 9, 9, 9, and 7, in which case there are 4 possible numbers: 7999, 9799, 9979 and 9997
Case ii: The 4 digits are 9, 9, 8, and 8, in which case there are 6 possible numbers: 8899, 8989, 8998, 9988, 9898, 9889
To the total number of 4-digit numbers in the box = 4 + 6 = 10

Aside: we could also you the MISSISSIPPI rule to determine the number of ways to arrange 9, 9, 8, and 8, but I wanted to see which numbers are divisible by 4.

Test for divisibility by 4: A number is divisible by 4 if and only it the 2-digit number created by the last 2 digits is divisible by 4.
So, since 88 is divisible by 4, we know that 9988 is divisible by 4.

Since 9988 is the only number divisible by 4, P(selected integer is divisible by 4) = 1/10

Answer: C