1. The problem statement, all variables and given/known data What is the number of positive integers n satisfying the property that the number b of positive integers a satisfying the property that a[tex]\leq[/tex]n(a is less than or equal to n)and n[tex]\leq[/tex]a^2(n is less than or equal to a^2) satisfies the property that the number of possible ways we can put 3 different objects into b different boxes is at least as big as the number of 0,1 sequences of length n. 2. Relevant equations 3 different objects into b different boxes = b^3 number of 0,1 sequences of length n = 2^n 3. The attempt at a solution 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.