Robust Unique Choices and Unique Neighbors

There are $n$ people in a line, and each independently and uniformly picks an integer from $\{1, 2, \dots, k\}$. A person is called unique if no other person picked the same number.

(a) Using indicator variables, find $E[U]$, the expected number of unique people.

(b) A unique neighbor pair is a pair of adjacent people $(i, i+1)$ who are both unique. Find $E[N]$, the expected number of unique neighbor pairs.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.