Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 47 / 4 / b / i / Solution

For fixed $p\ge 1$, the complement of the simple matching coefficient is the fraction of coordinates that disagree:This is the normalized Hamming distance. It is nonnegative, symmetric, and is zero exactly when every coordinate agrees. At each coordinate,Summing and dividing by $p$ proves the triangle inequality. Therefore $d_1$ is a metric on the binary vectors.