Three gnomes and three elves

[adm45]

Please explain when to use the Glue Method. In both WT edition 4 pg 73 (movie example) and pg 76 Q14, the questions says find number of combinations without something/one next to another? It is possible to use the Glue Method for Q14, correct? I understand how to use the Slot Method for this problem. I would like to know how to solve using the Glue Method or at least how to recognize that Glue Method will not be fastest method.

Thanks,

[jlucero]

Another instructor already answered your question in this thread. Glue method doesn't work here. Read her explanation and let us know if you have another question.

[adm45]

Ron,

Can you at least help here?

I
Jamie says the glue method is for when people/objects can't be next to each other. Then she goes on to say that the glue method would be way more difficult to use? can we use the glue method or not? if yes, how doe we quikcly recognize that we can't? I dont understand Jamie's explaination. If not' Why doesn't glue method work for gnomes and elves when we are told they can't be next to each other?

[jnelson0612]

Ron,

Can you at least help here?

I
Jamie says the glue method is for when people/objects can't be next to each other. Then she goes on to say that the glue method would be way more difficult to use? can we use the glue method or not? if yes, how doe we quikcly recognize that we can't? I dont understand Jamie's explaination. If not' Why doesn't glue method work for gnomes and elves when we are told they can't be next to each other?

Let me try to be more clear. In the context of seating, the glue method works well if all the members of the group except for a small minority of members (often two members) can be near each other. A great example of the use of the glue method is seen here:http://www.manhattangmat.com/forums/combination-manhattan-cat-5-problem-31-t3037.html

In this case, every member of the set has a restriction: every elf cannot sit next to another elf, and every gnome cannot sit next to another gnome. The glue method is not the way to solve a problem in which every element has a restriction. Much better to actually work out the possible seating scenarios.

[jasonthomasyee]

Ron,

Can you at least help here?

I
Jamie says the glue method is for when people/objects can't be next to each other. Then she goes on to say that the glue method would be way more difficult to use? can we use the glue method or not? if yes, how doe we quikcly recognize that we can't? I dont understand Jamie's explaination. If not' Why doesn't glue method work for gnomes and elves when we are told they can't be next to each other?

Let me try to be more clear. In the context of seating, the glue method works well if all the members of the group except for a small minority of members (often two members) can be near each other. A great example of the use of the glue method is seen here:http://www.manhattangmat.com/forums/combination-manhattan-cat-5-problem-31-t3037.html

In this case, every member of the set has a restriction: every elf cannot sit next to another elf, and every gnome cannot sit next to another gnome. The glue method is not the way to solve a problem in which every element has a restriction. Much better to actually work out the possible seating scenarios.

Hi guys,

I understand why the glue method is not optimal here -- it's much better just to use the slot method 6*3*2*2*1*1 = 72. Fair enough.

I am a little curious, though, how I would go about using the glue method here as I think it help me solidify my understanding of just how and why exactly the glue method works. I understand that the glue method is not ideal here because we have to consider scenarios where 3 gnomes are next to each other, 3 elfs are next to each other, 2 gnomes are next to each other, or 2 elfs are next to each other.

With 720 total combinations with no restrictions, my formula looked like:

720 - 3!4! - 3!4! - 2(5!) - 2(5!) = -68. Obviously wrong.

for reference, 3!4! comes from 3 elves or gnomes next to each other, and of course 2(5!) comes from 2 elves or gnomes next to each other. Where am I going wrong in this calculation?

[RonPurewal]

Well, first, you have a bunch of 3's and 4's, and/or 2's and 5's, suggesting that there are seven seats to be filled by seven people/animals/whatever.
There are only six seats, to be filled by six people/animals/entities.

Also, you're going to get a lot of overlap this way. If you just account for, e.g., one pair of gnomes sitting together and then use a factorial for everybody else, you're going to repeat a lot of cases that already appear elsewhere.
For instance, if those two gnomes happen to be next to the third gnome, you'll be repeating a case from the "3 gnomes together" calculation. If there are two elves together among the others, that case will be repeated, too. And so on.

In general, if an approach gets ridiculously complicated"”especially if there's another approach that is much less fraught (e.g., the slot method here)"”it's best to leave it, for the sake of developing and/or maintaining good timing habits.