There is a complicated mathematical way and there is a simple logical way to solve this problem. Perhaps there exist some other ways of solving it.
Let N be the total number of different persons we have. The number of persons (A, B, and C) we want them to be in one order is a = 3 here.1) Simple equipartition principle.
Let us say we have arranged all the N persons in all possible permutations. So that will be N! arrangements. Let us say that in each arrangement the person on the left side speaks before the person on his/her right side.All permutations of a = 3 number of persons (A, B, C) : = 3! = 6
A B C ; A C B ; B C A ; B A C ; C A B ; C B A
In the above, there is only one arrangement A B C - the first one, which satisfies the condition of A speaking before B and B speaking before C.
The N ! permutations
of all N persons will be equally divided in to the above six groups like the following. xx denotes some sequence of persons.
xx A xx B xx C xx ; xx A xx C xx B xx ;; xx C xx A xx B
xx B xx C xx A xx ; xx B xx A xx C xx ;; xx C xx B xx A So the number of arrangements we want are N ! / a! = 10! / 3!
=====================This can also be derived using permutations in another way.
Let the arrangements we want be in the following way:
Let N = 10. Let M = N - 3 = 7. Let there be m number of persons speaking before A, n number of persons between A and B, then p number of persons between B and C. Finally, after C there remain (M - m - n - p) number of persons to speak.
..m persons .. A ..n persons.. B .. p persons.. C .. (M - m - n - p) persons..
m varies from 0 to M. n varies from 0 to the remaining (M - m).
so p varies from 0 to (M - m - n).
For each value of p we have (M - m - n - p) ! permutations of arrangements after C. The number of ways of selecting p persons from available (M - m - n) number is:
P(M - m - n, p) for p = 0 , 1, 2, ... , (M - m - n)
So total arrangements are :
The above number of ways is the way of arranging persons after B. This is the number for each arrangement of n persons between A and B. n varies from 0 to M-m. The number of ways of selecting n persons is :
Let n' = M-m-n+1
Similarly: there are P(M, m) ways of selecting m members before A and for each selection the above calculated number is the number of ways others follow A for speaking. now m varies from 0 to M.
let m' = M-m+1
=================that is the answer...If you have N persons to speak and n persons have to speak in one order, then the answer will be : N ! / n !