On some Questions of Combination

Questions of combination are so connected with an actuary's business, that no apology is necessary for offering solutions to the notice of the readers of this Journal . When two systems, A and B, of combinations, each having its own individual cases, are so related that each case of A admits of being converted into a case of B, and into one only, by a rule which will not convert any other case of A into that same case of B, it may be said that the system A is convertibly within B. It is obvious that if A be convertibly within B, and B convertibly within A, the numbers of cases in the systems A and B are equal.