Using Line method for Dwarfes and Elves problem in Chapter4

[sanjay_s]

Hi,

I was trying to solves the dwarfes and elves problem from chapter4 Q 14 using line method, I couldn't quite figure out how to do it.

The question is:

Three dwarfes and three elves sit down in a row of six chairs. If no dward will sit next to another dwarf and no elf will sit next to another elf. In how many ways can the elves and dwarfes sit ?

From the line method It says :
no of ways (not sitting together)= total no of ways - total number of ways sitting together.

we have _ _ _ _ _ _

Now, no of ways of arranging all the elves and dwarfes is 6!

Now grouping all the dwarfes together and all the elves together we have two spaces ______ _______
no of ways is 2* (arrangements of dwarfes) * (no of ways to arrange elves) = 2*3!*3! =72

But, I know there are more combinations that haven't been accounted for when deducting from all combinations. I am hoping someone might be able to show how it is done. I can see that it can quickly get very tricky.

Bottomline: When do you use the line method and when do you use the anagram method ?

thanks for your response

[RonPurewal]

you can do this problem using the 'traditional' line method (one blank per seat), although it requires more than the usual amount of outside-the-box thinking.

to wit:

there are 6 ways to fill in the first line, since any of the 6 entities can sit in the first seat.
KEY REALIZATION: once you place the first entity, you have thus determined whether the seating is dedede or ededed (i.e., if you placed a dwarf, then it's dedede, and if you placed an elf, it's ededed)
therefore, there are 3 ways to fill in the second line: if you put a dwarf in the first spot, then you pick from the 3 elves, and if you put an elf in the first spot you pick from the 3 dwarves.

if you continue to apply the same line of reasoning, you get
6 * 3 * 2 * 2 * 1 * 1 = 72 arrangements, which is in agreement with other methods.

--

the line method is for problems that involve successive or independent choices. if you can classify a problem in this way, or rewrite or recast a problem so that it can be classified in this way, then the line method is applicable.
the anagram method is for problems involving the selection of groups from within larger groups. if you can classify a problem in this way, or rewrite or recast a problem so that it can be classified in this way, then the anagram method is applicable.

many problems can be re-cast in ways so that either method is applicable.

also, the anagram method is actually just a special case of the line method, with the added technique of dividing by factorials of redundant items. however, that's difficult for many students to see, so it helps to present them as 2 different methods. (even mathematicians tend to view them as such, dubbing the anagram method 'combinations/permutations' and the line method 'fundamental counting principle').

[sanjay_s]

Hi Ron,

Thanks for the response, It does really clear up some things in my mind. I also liked how you suggested strategies for when to with each of the methodologies.

thanks again
--Sanjay

[rfernandez]