C++, <10 lines, annotated


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    Every number that has 1 in a certain bit will contribute to increment of hamming distance when "paired" with every other number that has 0 in this bit. So look at every single bit exclusively, check how many numbers has it set to 1 and how many has 0, take a product and sum it over all 32 bits.

    int retval = 0;
    for(int bit=0; bit<32; bit++){
        int nOnes = 0, nZeros = 0;
        for(int i=0; i<nums.size(); i++)
            if( nums[i] & (1<<bit) ) nOnes++;
            else nZeros++;
        retval += nZeros * nOnes;
    }
    return retval

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