Question

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.)

Answer 1

There are 1680 ways to construct the octahedrons.

Here,

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron.

An octahedron has 8 equilateral triangles.

We have to find number of ways to construct the octahedron.

What is Regular octahedron?

An octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

Now,

An octahedron has 8 equilateral triangles.

We have to fix 2 faces before applying the colors.

So, the first face that can be fixed for any of the 8 faces but each faces are similar.

Hence, there are 8/8 ways.

And, When we fix one face, three adjoining faces are similar.

So, we have 7 colors for the second face.

Hence, 7/3 ways

Now, we can place any color anywhere, it will be different arrangement.

So, 6! ways.

Hence, Total number of ways = (8/8) * (7/3) * 6! = 7 * 6 * 5 * 4 *2 = 1680

So, There are 1680 ways to construct the octahedrons.

Learn more about the Regular octahedron visit:

https://brainly.com/question/11729152

#SPJ4