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

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10 00:00

Difficulty:   95% (hard)

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

HideShow timer Statistics 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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Math Expert V
Joined: 02 Aug 2009
Posts: 7981
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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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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Joined: 21 Nov 2018
Posts: 8
Location: India
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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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
Intern  B
Joined: 04 Feb 2019
Posts: 3
Re: Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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

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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
Manager  P
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Eight congruent equilateral triangles, each of a different color, are  [#permalink]

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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
Intern  B
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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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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+1 Kudos if you like the Question Re: Eight congruent equilateral triangles, each of a different color, are   [#permalink] 10 May 2019, 11:31
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