So I have attempted a solution... Does this make sense? The question is incredibly confusing wording but here are my thoughts to the problem...
3 different objects into b different boxes = b^3
number of 0,1 sequences of length n = 2^n
From the question, we know n is between a and a^2.
n is the number of positive integers
a is the positive integers before n
b is the number of a
and we know that b^3 is greater or equal to 2^n
if we try n=1, then a=1, b=1
n=2, then a=1,2, b=2
n=3, then a=1,2,3, b=3...
therefore n=b, which gives n^3 is greater than or equal to 2^n.
I don't know how to get n in other way, so I tried until n=10.
if n=10, then a=1,2,3,4,5,6,7,8,9,10, b=10
then 10^3 is not greater or equal to 2^10.
Therefore n is between 2 and 9 since if n=1, then a=1, b=1
1^3 is not greater of equal to 2^1.
Answer is 8.