At an election meeting 10 speakers are to address the meeting. The only protocol to be observed is that whenever they speak P.M. will speak before M.P., and M.P. will speak before M.L.A. In how many ways can the meeting be addressed?

Question

At an election meeting 10 speakers are to address the meeting. The only protocol to be observed is that whenever they speak P.M. will speak before M.P., and M.P. will speak before M.L.A. In how many ways can the meeting be addressed?

Vedantu Content Team · Accepted Answer

Hint:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                We start solving the problem by assuming the order P.M., M.P., and M.L.A., as a single person. We then recall the fact that the total number of ways to arrange n objects in n places is  $ n! $  ways. We make use of this fact for arranging the assumed person along with 7 others in the remaining places to proceed through the problem. We then make use of the fact that  $ n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times ......\times 2\times 1 $  to get the required answer.Complete step by step answer:According to the problem, we are given 10 speakers to address in an election meeting. We need to find the number of ways that the meeting can be addressed such that the only protocol is that whenever they speak P.M. will speak before M.P., and M.P. will speak before M.L.A.Let us assume that there is only one P.M., one M.P., and one M.L.A. while addressing the meeting.Let us assume that the order P.M, M.P, M.L.A as a single person and then arrange these resultant 8 persons in 8 places.We know that the number of ways to arrange n objects in n places is  $ n! $  ways.So, the meeting can be addressed in  $ 8! $  ways.We know that  $ n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times ......\times 2\times 1 $ . So, we have  $ 8!=8\times 7\times 6\times 5\times 4\times 3\times 2\times 1=40320 $ . $ \therefore $  We have found the number of ways to address the meeting as 40320 ways.Note: Here we have not given the preference for the P.M to address in the meeting as this will change the answer to the problem. Here we need to assume the data regarding the number of M.P’s and M.L.A’s attended to address the meeting as this is the important factor in finding the total number of ways. Similarly, we can expect the problems to find the total number of ways of addressing the meeting if P.M addresses last of all.