We consider a transmitter broadcasting random linear combinations (with fieldsize $d$) formed from a block of $c$ packets to $n$ (heterogeneous) receivers, over independent erasure channels. We establish several properties of the random delay until all $n$ receivers have recovered all $c$ packets, denoted $Y_{n:n}^{(c)}$.  First, we provide tight bounds, exact expressions, and a recurrence for the moments of $Y_{n:n}^{(c)}$.  Second, we study the delay per packet $Y_{n:n}^{(c)}/c$ as a function of $c$, and its monotonicity properties.  Third, we employ extreme value theory to investigate $Y_{n:n}^{(c)}$ as a function of $n$ (as $n to infty$). Several results are new, some are extensions of existing ones, and some are proofs of known results using new (probabilistic) proof techniques.