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How many ways are there to write 2000 as the sum of powers of 2 with non-negative exponents, where each power is used at most twice?

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    Divide by two until the remainder is 0 or 1.
    The corresponding quotients will give the powers of 2 to make up 2000.

    2000/2=1000 R0
    1000/2=500 R0
    500/2=250 R0
    250/2=125 R0
    125/2=62 R1
    62/2=31 R0
    31/2=15 R1
    15/2=7 R1
    7/2=3 R1
    3/2=1 R1
    So 2000 in binary notation is
    11111010000
    or
    the powers are
    4,6,7,8,9,10,11
    or
    2^4+2^6+2^7+2^8+2^9+2^10
    =2000
    There is only one way if the powers are not to be repeated.

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