Permutations of (abc)(efg)(h) in S7

[cragar]

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.

 

[Samy_A]

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.

Is the permutation (abc)(efg)(h) different from the permutation (efg)(abc)(h)?

 

[cragar]

no, so I then need to divide by 3!

 

[haruspex]

no, so I then need to divide by 3!

How do you arrive at 3?

 

[cragar]

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.

 

[haruspex]

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, I understood it was factorial, but I was specifically challenging the 3.
I only see two equivalent arrangements there. What other four have you redundantly counted by your method?

 

[cragar]

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.

 

[haruspex]

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.

Yes, 2, but not for that reason.
You chose abc as a set of three, so you covered all permutations of those three in one selection - no double counting so far.
Likewise in choosing the set def.
The double counting arises because you could have chosen the set def first, then the set abc.