The problem involves determining the number of distinct permutations of the form (abc)(efg)(h) within the symmetric group S7, which consists of 7 elements. The discussion centers around the combinatorial aspects of selecting and arranging these elements.
The discussion is active, with participants exploring different interpretations of how to count the arrangements and questioning the assumptions behind their calculations. Some guidance has been offered regarding the factorials involved, but no consensus has been reached on the exact counting method.
Participants are navigating the complexities of permutations and combinations, particularly in relation to the redundancy in counting arrangements of the cycles. There is an ongoing debate about the correct approach to avoid overcounting.
Is the permutation (abc)(efg)(h) different from the permutation (efg)(abc)(h)?cragar said:Homework Statement
How many distinct permutations are there of the form (abc)(efg)(h) in S7?
Homework Equations
3. The Attempt at a Solution [/B]
since we have 7 elements I think for the first part it should be 7 choose 3 then 4 choose 3.
And then we multiply those together.
How do you arrive at 3?cragar said:no, so I then need to divide by 3!
Yes, I understood it was factorial, but I was specifically challenging the 3.cragar said:i mean 3 factorial, because (abc)(efg)(h)= (efg)(abc)(h) so I need to divide by 6 because their are 6 ways to arrange that and we would be over counting.
Yes, 2, but not for that reason.cragar said:oh your saying only divide by 2! because by the time we choose our last element h we only have one choice, so it won't affect the counting.
Because by the time (abc)(efg) have been chosen we only have one choice left.