Messaging probabilities I am part of a large family - we have twenty-four people who send texts back and forth, in various configurations. What would be the total number of possible message threads? All the different one-on-ones - the three persons groups, four, etc., up to all twenty-four of us on one thread.

If we encode:
$(0110\cdots)$ as 1st member (alphabetical ordering say) not included, 2nd and 3rd included, 4th not, so forth...
Then there are $2^{24}-25$ ways.
$2^{24}$ is the number of such binary strings, and 24 of those are messages to oneself, which we discount, as well as empty threads (messages to no one)

It is worth noting that the sums of  nCr and 2n are related, in fact:
  $$\sum^n_{i=0}\ ^nC_i=2^n$$
  So what this answer has is:
  $\sum^{24}_{n=2}\ ^{24}C_n=\sum^{24}_{n=0}\ ^{24}C_n -\ ^{24}C_0 - \ ^{24}C_1=2^{24}−1−24=2^{24}−25$ - as I claimed.

