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# Eight congruent equilateral triangles, each of a different color, are

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Math Expert
Joined: 02 Sep 2009
Posts: 58453
Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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18 Mar 2019, 01:58
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Difficulty:

95% (hard)

Question Stats:

20% (01:47) correct 80% (02:23) wrong based on 40 sessions

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Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

(A) 210
(B) 560
(C) 840
(D) 1260
(E)}1680

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Joined: 02 Aug 2009
Posts: 7981
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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18 Mar 2019, 05:33
Bunuel wrote:

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

(A) 210
(B) 560
(C) 840
(D) 1260
(E)}1680

An octahedron has 8 triangles.
So, we have to fix 2 faces, before applying colours.
So, the first face that can be fixed can be any of the 8 faces but each face will be similar , so 8/8 ways.
Now, when you have fixed one face, the three adjoining faces are similar, so we have 7 colours for the second face, so 7/3.
Now, you can place any color anywhere, it will be a different arrangement, so 6!
Total = (8/8)*(7/3)*6!=1*7*6*5*4*2=1680

E
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Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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20 Mar 2019, 07:05
chetan2u wrote:
Bunuel wrote:

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

(A) 210
(B) 560
(C) 840
(D) 1260
(E)}1680

An octahedron has 8 triangles.
So, we have to fix 2 faces, before applying colours.
So, the first face that can be fixed can be any of the 8 faces but each face will be similar , so 8/8 ways.

Now, when you have fixed one face, the three adjoining faces are similar, so we have 7 colours for the second face, so 7/3.
Now, you can place any color anywhere, it will be a different arrangement, so 6!
Total = (8/8)*(7/3)*6!=1*7*6*5*4*2=1680

E

PLEASE EXPLAIN WHAT YOU MEAN BY FIX 2 FACES
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Joined: 04 Feb 2019
Posts: 3
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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20 Mar 2019, 07:14
chetan2u, Bunuel, VeritaKarishma please explain what do you mean by 7/3 ?
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Joined: 12 Sep 2018
Posts: 27
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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24 Mar 2019, 05:17
Divide the figure into 2 pyramids. Each will have a unique combination of 4 colours.
The number of these combinations is 8C4=70.
Then, considering the 4 faces of the pyramid, we need to find the number of different arrangements of the colour, hence the permutation 4!=24.

Finally 8C4x4!=70x24=1680 E
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Joined: 01 Feb 2017
Posts: 242
Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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28 Mar 2019, 04:07
1
Total number of faces= 8
Total number of colors= 8
Now, 8 colors can be placed in these 8 slots in 8! ways.

Among these 8! combinations, there are 24 sets of mirror images:
As octahedron has 6 vertices, there are 6 options for a vertices to be placed on.
And for each position of vertices, there are 4 rotations available.
Hence, there are 6*4 = 24 sets of mirror images.

Therefore, Distinguishable Octahedrons (i.e total combinations corrected for mirror images) = 8!/24= 1680.
Ans E
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Joined: 24 Dec 2018
Posts: 33
GMAT 1: 740 Q50 V40
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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10 May 2019, 11:31
Shobhit7 wrote:
Total number of faces= 8
Total number of colors= 8
Now, 8 colors can be placed in these 8 slots in 8! ways.

Among these 8! combinations, there are 24 sets of mirror images:
As octahedron has 6 vertices, there are 6 options for a vertices to be placed on.
And for each position of vertices, there are 4 rotations available.
Hence, there are 6*4 = 24 sets of mirror images.

Therefore, Distinguishable Octahedrons (i.e total combinations corrected for mirror images) = 8!/24= 1680.
Ans E

Can you please elaborate how did you came up with number of mirror images (may be by considering other 3-D figure).
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Re: Eight congruent equilateral triangles, each of a different color, are   [#permalink] 10 May 2019, 11:31
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