Three dwarves and three elves sit down in a row of six

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Three dwarves and three elves sit down in a row of six [#permalink]

16 Feb 2011, 13:13

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Three dwarves and three elves sit down in a row of six chairs. If no dwarf will sit next to another dwarf and no elf wil sit next to another elf, in how many different ways can the elves and dwarves sit?

as per my solution

on first chair from 3dwarves any one can sit on another chair from 3elves any one can sit nd so on

3*3*2*2*1*1=36

But OA is

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Last edited by Bunuel on 28 Feb 2012, 14:07, edited 2 times in total.

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Re: arrangements made [#permalink]

16 Feb 2011, 13:27

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Three dwarves and three elves sit down in a row of six chairs. If no dwarf will sit next to another dwarf and no elf wil sit next to another elf, in how many different ways can the elves and dwarves sit?

In order to meet the restriction dwarves and elves must sit either DEDEDE or EDEDED. There are 3!*3!=36 arrangements possible for each case (3! arrangements of dwarves and 3! arrangements of elves), so total ways to sit are 2*36=72.
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Re: arrangements made [#permalink]

17 Feb 2011, 06:29

GMATD11 wrote:

Pls chk the image

as per my solution

on first chair from 3dwarves any one can sit on another chair from 3elves any one can sit nd so on

3*3*2*2*1*1=36

But OA is 72

You forgot to multiply the answer with 2. You see, either one the dwarves can take the first seat or one of the elves can take it. If a dwarf occupies the first seat, the other dwarves will sit on the 3rd and 5th seat. Elves will sit on 2nd, 4th and 6th seats. So as you have rightly calculated above, there will be 36 seating arrangements.

In the event where an elf takes the first seat, there will be as many combinations possible albeit the seats will change, with the elves taking the odd numbered seats and the dwarves taking the even numbered ones.

So total number of combinations= 36+36= 72.
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elves [#permalink]

28 Feb 2012, 13:57

there are only two patterns: dedede, ededed. each will return same number of combinations.

combinations = 3*3*2*2*1*1*(2) = 72 ... multiply (2) because there are two patterns
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combination [#permalink]

05 Apr 2012, 12:51

hello

IF 3 DWARFS AND THREE ELVES SIT DOWN IN A ROW OF 6 chairs . IF NO DWARFS WILL SIT NEXT TO ANOTHER DWARFS AND NO ELFS WILL SEAT NEXT TO ANOTHER ELFS
IN HOW MANY WAYS CAN THE ELVES AND DWARFS SIT?

answer to this practice exercises is 72
please explain your reasoning

thanks

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Re: combination [#permalink]

05 Apr 2012, 13:09

keiraria wrote:

Merging similar topics. Please ask if anything remains unclear.

P.S. Please turn Caps Lock off when posting.
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Re: combination [#permalink]

05 Apr 2012, 13:33

Bunuel wrote:

keiraria wrote:

Merging similar topics. Please ask if anything remains unclear.

P.S. Please turn Caps Lock off when posting.

Thanks my approach was the total number of way the 6 could sit minus the number of way the dwarf and the elfs can sit together
6!-3!x3!

i dont understand my mistake
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Re: combination [#permalink]

05 Apr 2012, 16:57

keiraria wrote:

Bunuel wrote:

keiraria wrote:

Merging similar topics. Please ask if anything remains unclear.

P.S. Please turn Caps Lock off when posting.

Thanks my approach was the total number of way the 6 could sit minus the number of way the dwarf and the elfs can sit together
6!-3!x3!

i dont understand my mistake
best regards

The problem with that solution is that 3!*3! does not give you all cases when dwarves and elves sit together. 3!*3! counts only the cases when all 3 dwarves and all 3 elves are sitting together for example all cases for DDDEEE. But since we are asked to find the ways to arrange them so that no dwarf will sit next to another dwarf and no elf wil sit next to another elf then there are many other cases possible when this condition is violated: EEEDDD, EEDDED, DDEEDE, ...
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Three dwarves and three elves sit down in a row of six chairs. I [#permalink]

07 Jan 2013, 11:51

ededed OR dedede
3*3*2*2*1*1 + 3*3*2*2*1*1
= 72

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Re: 3 dwarves and 3 Elves sit down in a row of 6 chairs. If no [#permalink]

07 Jan 2013, 11:56

you can either have the following arragements of seating positions
EDEDED or DEDEDE

Therefore for the first arragement you would have 3!x3! ways to sit the 6 people. Since there are two possible seating arragememts you would have 2(3!x3!) = 72

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Three dwarves and three elves sit down in a row of six chairs. I [#permalink]

22 Jan 2013, 03:34

How would we do this using COMBINATIONS/C as opposed to permutations or just counting out possibilities for each of the 6 slots (ie 6x3x2x2)?
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Re: Three dwarves and three elves sit down in a row of six chair [#permalink]

22 Jan 2013, 21:17

manimgoindowndown wrote:

How would we do this using COMBINATIONS/C as opposed to permutations or just counting out possibilities for each of the 6 slots (ie 6x3x2x2)?

You need to arrange the 6 people in 6 places. You can think in terms of combinations in the following way:

For the first chair, we select any one person of the 6 in 6C1 ways.
For the second chair, we select any one from the 3 in 3C1 ways (e.g. if an elf is sitting on the first chair, we need to choose out of the 3 dwarves only)
For the third chair, we select one of the leftover 2 (e.g. one elf is already sitting in first chair) in 2C1 ways
For the fourth chair, we select one of the leftover 2 (e.g. one dwarf is already sitting in the second chair and we need to choose of the remaining 2) in 2C1 ways
For the fifth and sixth chairs, there is only one choice each.

Answer: 6C1 * 3C1 * 2C1 * 2C1 = 6*3*2*2 = 72
(note that this is the basic counting principle itself. For more, check: http://www.veritasprep.com/blog/2011/10 ... inatorics/)
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Re: Three dwarves and three elves sit down in a row of six chair [#permalink] 22 Jan 2013, 21:17

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