A permutation is often represented by the cycles it has. For example, if we permute the numbers in the natural order to 2 3 1 5 4, this is represented as (1 3 2) (5 4). In this the (132) says that the first number has gone to the position 3, the third number has gone to the position 2, and the second number has gone to position 1, and (5 4) means that the fifth number has gone to position 4 and the fourth number has gone to position 5. The numbers with brackets are to be read cyclically.
If a number has not changed position, it is kept as a single cycle. Thus 5 2 1 3 4 is represented as (1345)(2).
We may apply permutations on itself If we apply the permutation (132)(54) once, we get 2 3 1
5 4. If we apply it again, we get 3 1 2 4 5 , or (1 2 3)(4) (5)
If we consider the permutation of 7 numbers (1457)(263), what is its order (how many times must it be applied before the numbers appear in their original order)?

• here we can see in the first cycle that is (1457) we will get our original order after exactly 4 cycles. hence whatever the answer it must be completely divisible by 4.
  ex: for 1: (1st cycle)=>1 goes to 4th location ,(2nd cycle)=>1 goes to 5th loc, (3rd cycle)=> goes to 7th loc (4th cycle)=> returns to 1st location, Same goes with rest three elements in this cycle
  
  Next is (263) we will get original order after 3 cycles here hence after ever 3 cycle our original order comes for ex after 3rd cycle and after 6th cycle and after 9th cycle. Hence our answer must be divisible by 3 also.
  
  Now we know answer must be divisible by 4 and 3 and we need our answer minimum. hence LCM of 3 and 4 will work. i.e. Answer is 12
• 7 years agoHelpfull: Yes(3) No(0)
• 3; after 1p-1 is in 5th place,2p-1 is in 7th place,3rd permutation,1is to 1st place so as remaining numbers
• 7 years agoHelpfull: Yes(2) No(0)
• as itis given permutation of (123)(54) is 5! so (1457)(263) has seven digit so its answer is 7!
• 7 years agoHelpfull: Yes(0) No(7)
• 4!×3! =144 ways we
  Once get original
• 5 years agoHelpfull: Yes(0) No(1)