Seating indistinguishable twins

Havie

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Given this question,
the part im hung up on is the "can not be distinguished part"

I would think you would take
14 total
and split into

triplets = 6 ppl
twins = 6 ppl
individuals = 2 ppl

and since they cant be distinguished from one another you would just do
3!
but thats not the answer.
what you actually have to do is
6! * 6! * 2! = 1,036,800

the part im having trouble wrapping my head around is what exactly this means if acted out.

is it saying
group 1 : can arrange 6 people 720 ways
group 2 : can arrange 6 people 720 ways
group 3 : can arrange 2 people 2 ways

what i picture this meaning is :
first 6 chairs could go 720 different ways ( all triplets in chair 1-6)
next 6 chairs could go 720 different ways ( all twins in chair 7-12)
last 2 chairs could go 2 different ways ( individuals in chair 13, 14)

is this not accounting for when they move out of order? like a twin sits in chair 2 and a individual sits in chair 5 like wise theres a triplet in chair 9 and 13?
or is that exactly what the multiplication is doing?
you have 720 different choices * 720 different choices * 2 choices?
I guess i dont see how this is mixing them together. and to what end? why is the answer not just 14! then?

if it was addition, 720+720+2 that would keep the people in their certain sections?

i feel like imthis close to understanding this but cant quite grasp it

any insight would be greatly appreciated, thanks


also -- holy **** this doesnt make any sense,
if there are "two sets of identical triplets, and three sets of identical twins" that are indistinguishable...... that makes zero sense because 2 sets or identical triplets couldn't possibly be indistinguishable, 3 triplets happen to look EXACTLY like 3 other triplets? isnt that genetically impossible?
I dont want to spark some political incorrectness but wouldn't this be easier to think of as like...
blue people, green people and red people? idk...
 
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Given this question,
the part im hung up on is the "can not be distinguished part"
A many a textbook calls this the "MISSISSIPPI" problem.
How ways can you rearrange that word? If it were \(\displaystyle MI_1S_1S_2I_2S_3S_4I_3P_1P_2I_4\) it is no problem.
WHY? Because now the subscripts make eleven different letters. In that case the answer is \(\displaystyle 11!\)
BUT for the string \(\displaystyle SSSS\) there is only one way to rearrange; while for the string \(\displaystyle S_1S_2S_3S_4\) there are \(\displaystyle 4!=24\) ways to rearrange it.
So in the \(\displaystyle 11!\) ways of arrangeing \(\displaystyle MI_1S_1S_2I_2S_3S_4I_3P_1P_2I_4\) by dropping the subscrips off the S's we have 24 arrangements identical.
So now there are \(\displaystyle \dfrac{11!}{4!\cdot 4!\cdot 2!}\) ways to rearrange the word MISSISSIPPI.

Lets apply that to your question.
Model it with fourteen letters: AAABBBCCDDEEFG. The A's & B's are the triplets; the C's, D's & E"s are twins; and F & G are individuals.
Applying the "MISSISSIPPI" counting principle what is your answer?
 
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this doesnt make any sense,
if there are "two sets of identical triplets, and three sets of identical twins" that are indistinguishable...... that makes zero sense because 2 sets or identical triplets couldn't possibly be indistinguishable, 3 triplets happen to look EXACTLY like 3 other triplets? isnt that genetically impossible?
I dont want to spark some political incorrectness but wouldn't this be easier to think of as like...
blue people, green people and red people? idk...

There is some ambiguity, but I think you are clearly intended to take EACH set of twins or triplets, separately, as indistinguishable, as (more or less) in real life. So, as was said, it's like arranging the letters AAABBBCCDDEEFG, where AAA and BBB are the sets of triplets, CC, DD, and EE are the sets of twins, and F and G are the singlets. They aren't saying that all triplets are alike; it's the identical ones (those in any given set) who are indistinguishable.
 
A many a textbook calls this the "MISSISSIPPI" problem.
How ways can you rearrange that word? If it were \(\displaystyle MI_1S_1S_2I_2S_3S_4I_3P_1P_2I_4\) it is no problem.
WHY? Because now the subscripts make eleven different letters. In that case the answer is \(\displaystyle 11!\)
BUT for the string \(\displaystyle SSSS\) there is only one way to rearrange; while for the string \(\displaystyle S_1S_2S_3S_4\) there are \(\displaystyle 4!=24\) ways to rearrange it.
So in the \(\displaystyle 11!\) ways of arrangeing \(\displaystyle MI_1S_1S_2I_2S_3S_4I_3P_1P_2I_4\) by dropping the subscrips off the S's we have 24 arrangements identical.
So now there are \(\displaystyle \dfrac{11!}{4!\cdot 4!\cdot 2!}\) ways to rearrange the word MISSISSIPPI.

Lets apply that to your question.
Model it with fourteen letters: AAABBBCCDDEEFG. The A's & B's are the triplets; the C's, D's & E"s are twins; and F & G are individuals.
Applying the "MISSISSIPPI" counting principle what is your answer?

so..\(\displaystyle \dfrac{14!}{3!\cdot 3!\cdot 2!\cdot 2!\cdot 2!}\) ?
this doesnt give me 1,036,800

neither does
\(\displaystyle \dfrac{14!}{6!\cdot 6!\cdot 2!}\) ?


There is some ambiguity, but I think you are clearly intended to take EACH set of twins or triplets, separately, as indistinguishable, as (more or less) in real life. So, as was said, it's like arranging the letters AAABBBCCDDEEFG, where AAA and BBB are the sets of triplets, CC, DD, and EE are the sets of twins, and F and G are the singlets. They aren't saying that all triplets are alike; it's the identical ones (those in any given set) who are indistinguishable.

Hmm if you separate them like that
isnt it
3! * 3! * 2! * 2! * 2! * 1! * 1 = 288?

the answer is 1,036,800
which can be reached by
6! * 6! *2!
which means they are grouping the sets of triplets and sets of twins as indistinguishable,
this means that , in real world terms that most ppl could know:

That the Olsen Twins (mary-kate, and ashley), would look IDENTICAL to another set of twins born at a later date, and also identical to ANOTHER set of twins born at a later date. thats not even genetically possible? So I dont understand why were asked to group them this way because i cant imagine it in the way they are saying, or the method behind what they are doing.
 
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so..\(\displaystyle \dfrac{14!}{3!\cdot 3!\cdot 2!\cdot 2!\cdot 2!}\) ?
this doesnt give me 1,036,800

neither does
\(\displaystyle \dfrac{14!}{6!\cdot 6!\cdot 2!}\) ?

Hmm if you separate them like that
isnt it
3! * 3! * 2! * 2! * 2! * 1! * 1 = 288?

the answer is 1,036,800
which can be reached by
6! * 6! *2!
which means they are grouping the sets of triplets and sets of twins as indistinguishable,
this means that , in real world terms that most ppl could know:

That the Olsen Twins (mary-kate, and ashley), would look IDENTICAL to another set of twins born at a later date, and also identical to ANOTHER set of twins born at a later date. thats not even genetically possible? So I dont understand why were asked to group them this way because i cant imagine it in the way they are saying, or the method behind what they are doing.

I had missed the fact that you claimed that 1,036,800 is the correct answer, not just what you thought it was.

If it isn't clear, we agree with you: That answer is WRONG.

It's either a very poorly written problem, or just a wrong answer. Answers in the back of books are not always correct!

Incidentally, I happen to be an identical twin myself; I'm somewhat offended even by claiming that a pair of twins are indistinguishable ...
 
I had missed the fact that you claimed that 1,036,800 is the correct answer, not just what you thought it was.

If it isn't clear, we agree with you: That answer is WRONG.

It's either a very poorly written problem, or just a wrong answer. Answers in the back of books are not always correct!

Incidentally, I happen to be an identical twin myself; I'm somewhat offended even by claiming that a pair of twins are indistinguishable ...
hmm okay,
thanks !
 
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