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.
A permutation of a group is a rearrangement of the elements in a group. It is a way of changing the order of the elements without changing the group itself.
The number of permutations in a group is equal to the factorial of the number of elements in the group. For example, if a group has 5 elements, there are 5! = 5 x 4 x 3 x 2 x 1 = 120 permutations.
If a group has n elements, with k1 elements of one type, k2 elements of another type, and so on, the number of permutations is given by n! / (k1! * k2! * ... * kn!). This formula takes into account that not all permutations will be unique due to the repeating elements.
A permutation involves rearranging the elements in a group, while a combination does not consider the order of the elements. For example, the permutations of the letters "ABC" would include "ABC", "ACB", "BAC", etc. while the combinations would only include "ABC", "ACB", and "BAC".
Permutations have many practical applications, such as in coding and cryptography, where they are used to generate unique combinations of characters. They are also used in statistics to calculate the number of possible outcomes in a sample space. Additionally, permutations are used in various fields of science, such as genetics, to analyze the different ways in which genes can be arranged.