It's a very nice proof. Note that this implicitly relies on a great trick used in probability theory, namely that E(X_1 + X_2 + ... + X_n) = E(X_1) + E(X_2) + .

It's a very nice proof. Note that this implicitly relies on a great trick used in probability theory, namely that E(X_1 + X_2 + ... + X_n) = E(X_1) + E(X_2) + ... + E(X_n), even if the random variables are not independent. In this specific case X_i = "the number of clumps where the ith slowest car is in front of" which is either 0 or 1.