This paper considers a massive connectivity setting in which a base-station (BS) aims to communicate sources (X1, · · · ; Xk;) to a randomly activated subset of k users, among a large pool of n users, via a common message in the downlink. Although the identities of the k active users are assumed to be known at the BS, each active user only knows whether itself is active and does not know the identities of the other active users. A naive coding strategy is to transmit the sources alongside the identities of the users for which the source information is intended. This requires H(X1, · · · ; Xk;) + k log(n) bits, because the cost of specifying the identity of one out of n users is log(n) bits. For large n, this overhead can be significant. This paper shows that it is possible to develop coding techniques that eliminate the dependency of the overhead on n, if the source distribution follows certain symmetry. Specifically, if the source distribution is independently and identically distributed (i.i.d.) then the overhead can be reduced to at most O(log(k)) bits, and in case of uniform i.i.d. sources, the overhead can be further reduced to O(1) bits. For sources that follow a more general exchangeable distribution, the overhead is at most O(k) bits, and in case of finite-alphabet exchangeable sources, the overhead can be further reduced to O(log(k)) bits. The downlink massive random access problem is closely connected to the study of finite exchangeable sequences. The proposed coding strategy allows bounds on the Kullback-Leibler (KL) divergence between finite exchangeable distributions and i.i.d. mixture distributions to be developed and gives a new KL divergence version of the finite de Finetti theorem, which is scaling optimal.