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Combinatorics(word question)

  1. Mar 16, 2010 #1
    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.
     
  2. jcsd
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