This is a modified version of the coupon collector problem, where we are interested in making comparisons between the "coupons". There are a number of constraints which have been placed in order to make this applicable to the application of interest (not relevant here, but related to clustering).

1. There exists $M$ unique coupons.
2. Coupons come in packets of size $K$.
3. Each packet contains $K$ unique units, sampled uniformly without replacement from the total set of $M$ units.
4. The contents of a packet is independent of all other packets.
5. All units in a packet are "compared" to all other units in the packet.
6. Units may not be compared across packets.

Question 1. Let $X$ be the number of unique comparisons that have been made after $T$ packets have been acquired. What is the expected value and variance of $X$?

Question 2: Let $T_\star$ be the smallest number of packets required to make all of the $\binom{M}{2}$ comparisons. What is the expected value and variance of $T_\star$?