Permutations Help: Solving Questions About Collect and Seating Arrangements

[adeel]

I need help with these two questions:

How many permutations are there of the word collect if the 2 l's have to be together and the two c's have to be separated?

I got as far as 360 because if u keep the l's together you would get 6! x 2!
2! x 2!

but after that I am stuck on what to do next

and the other question:

How many ways can 4 couples be seated at a circular table so that the couples are never sitting together?

i knoe 8! / 8 gives the possibilities of all of them at one table, but since some of them can sit together i don't know what to do

can someone help me finish of these questions?

 

[Hurkyl]

So you know how to find the number of combinations where both l's are together.

Can you find how many combinations there are when both l's and both c's are together?

Can you guess where to go next?



I think you can do the other problem with a variation on this trick.

 

[adeel]

hmm

so what u are saying is figure out how many ways u can have just the l's together, and then figure out how many ways u can have both c's and l's together and subtract to get how many ways l can be together but c is not together?

so 360 (what i got earlier) minus (5! x 2! x 2! / 2! x 2! which gives an answer of 120)

so 360 - 120 = 240 ways? Do you know that is the right way to do it or you think?

Can u explain how the variation of the second question would work?

 

[Hurkyl]

Your answer for "collect" looks right.


For the second question... there might be a simpler way, but off hand I see this one:

Find the number of ways the couples can have a seat where all couples sit together.

Find the number of ways they can have a seat where three pairs sit together and the other 2 can be anywhere.

Same for two pairs.

Same for one pair.

and then subtract from the number of arrangements.