Let n e N. Explain what is meant by saying that Pi is a permutation of n = {1, 2, ..., n}.
Now let n = 6. How many permutations are there of 6?

The permutation Pi of 6 is given below in two line notation. Write Pi in disjoint cycle notation.

1 2 3 4 5 6
1 5 6 3 2 4

How many permutations of 6 are there of the form (a b c) (d e f) when written in disjoint cycle
notation?

got Pi written in disjoint cycle notation as
(2 5)(3 6 4), but can't do the other parts.

Question

Let n e N. Explain what is meant by saying that Pi is a permutation of n = {1, 2, ..., n}.
Now let n = 6. How many permutations are there of 6?

The permutation Pi of 6 is given below in two line notation. Write Pi in disjoint cycle notation.

1 2 3 4 5 6
1 5 6 3 2 4

How many permutations of 6 are there of the form (a b c) (d e f) when written in disjoint cycle
notation?

got Pi written in disjoint cycle notation as
 (2 5)(3 6 4), but can't do the other parts.

anonymous · Answer

first answer is 6!

anonymous · Answer

how? please explain

anonymous · Answer

ok.  you have 6 elements to permute.  call them {1,2,3,4,5,6}

anonymous · Answer

you have 6 choices of where to send 1.   then you have 5 choices of where to send 2, for a total of 6*5 choices.  then 4 choices of where to send 3 etc. so all together you have 6*5*4*3*2*1 choices

anonymous · Answer

Ok so the answer is 720? thanks. do you know how to do other parts?

anonymous · Answer

i am thinking of a nice theorem that gives it, but it is not coming to me.  so maybe we can just count.

anonymous · Answer

for the last part?

anonymous · Answer

i am going to say 80 but i could be wrong.

anonymous · Answer

sorry, how did you get 80?

anonymous · Answer

reasoning as follows. for (a,b,c) we have 6 choices for a, then 5 for b and 3 for c but the permutation (abc) is the same as (bca) same as (cab) so we have over counted by a factor of 3, and so for the first cycle there are 6*5*4/3=40 ways.  then for the next there are only actually two possibilities. say we are left with {def} then the only three cycles are (def) and (dfe) and 4082=80

anonymous · Answer

last line should be 40*2=80

anonymous · Answer

let me think for a moment to see if this is correct.

anonymous · Answer

4 for C right? and thanks. okay.

anonymous · Answer

i get the first part, the first cycle but no idea about the second.

anonymous · Answer

i really want to say there are 
$$\dbinom{6}{3}\times \dbinom{3}{2}=20\times 3=60$$ for C.

anonymous · Answer

okay i get the first part, but why is it 3C2? 

answer is 60 not 80?

anonymous · Answer

because i am wrong. answer to C is 80. 80 different 3 cycles in S6 http://www.math.unl.edu/~bharbourne1/M417Spr03/M417Hmwk7Sols.html

anonymous · Answer

thanks. i understand it until 5(4)(2)(1) = 40. where im confused as to where the numbers come from. i get that they are b,c,e,f but dont understand why we use those numbers.