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Pólya Enumeration Theorem

Description: The Pólya Enumeration Theorem is a fundamental result in combinatorics that establishes a connection between the number of symmetries of an object and the number of ways it can be colored. This quiz will test your understanding of the Pólya Enumeration Theorem and its applications.
Number of Questions: 16
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Tags: combinatorics pólya enumeration theorem symmetry coloring
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What is the Pólya Enumeration Theorem?

  1. A theorem that relates the number of symmetries of an object to the number of ways it can be colored.

  2. A theorem that relates the number of permutations of a set to the number of ways it can be partitioned.

  3. A theorem that relates the number of combinations of a set to the number of ways it can be ordered.

  4. A theorem that relates the number of subsets of a set to the number of ways it can be arranged.

Show answer
Correct Option: A
Explanation:

The Pólya Enumeration Theorem states that the number of ways to color an object with $n$ colors is equal to the sum of the number of orbits of the symmetry group of the object acting on the set of $n$-colorings.

What is a symmetry group?

  1. A group of transformations that preserve the structure of an object.

  2. A group of transformations that change the structure of an object.

  3. A group of transformations that move an object from one place to another.

  4. A group of transformations that rotate an object around an axis.

Show answer
Correct Option: A
Explanation:

A symmetry group is a group of transformations that preserve the structure of an object. In other words, if you apply a symmetry transformation to an object, the object will look exactly the same as before.

What is an orbit of a symmetry group?

  1. The set of all points that are moved by a symmetry transformation.

  2. The set of all points that are fixed by a symmetry transformation.

  3. The set of all points that are colored by a symmetry transformation.

  4. The set of all points that are not colored by a symmetry transformation.

Show answer
Correct Option: A
Explanation:

The orbit of a symmetry group is the set of all points that are moved by a symmetry transformation. In other words, it is the set of all points that are equivalent under the symmetry group.

How many ways can a cube be colored with $6$ colors if each face must be colored with a different color?

  1. $6$

  2. $6!$

  3. $6^6$

  4. $6^{24}$

Show answer
Correct Option: D
Explanation:

A cube has $6$ faces, so there are $6^{24}$ ways to color it with $6$ colors if each face must be colored with a different color.

How many ways can a tetrahedron be colored with $4$ colors if each face must be colored with a different color?

  1. $4$

  2. $4!$

  3. $4^4$

  4. $4^{12}$

Show answer
Correct Option: D
Explanation:

A tetrahedron has $4$ faces, so there are $4^{12}$ ways to color it with $4$ colors if each face must be colored with a different color.

How many ways can a regular octahedron be colored with $8$ colors if each face must be colored with a different color?

  1. $8$

  2. $8!$

  3. $8^8$

  4. $8^{48}$

Show answer
Correct Option: D
Explanation:

A regular octahedron has $8$ faces, so there are $8^{48}$ ways to color it with $8$ colors if each face must be colored with a different color.

How many ways can a regular dodecahedron be colored with $12$ colors if each face must be colored with a different color?

  1. $12$

  2. $12!$

  3. $12^{12}$

  4. $12^{120}$

Show answer
Correct Option: D
Explanation:

A regular dodecahedron has $12$ faces, so there are $12^{120}$ ways to color it with $12$ colors if each face must be colored with a different color.

What is the Burnside's Lemma?

  1. A lemma that relates the number of orbits of a symmetry group to the number of ways an object can be colored.

  2. A lemma that relates the number of permutations of a set to the number of ways it can be partitioned.

  3. A lemma that relates the number of combinations of a set to the number of ways it can be ordered.

  4. A lemma that relates the number of subsets of a set to the number of ways it can be arranged.

Show answer
Correct Option: A
Explanation:

Burnside's Lemma states that the number of ways to color an object with $n$ colors is equal to the average number of fixed points of the symmetry group of the object acting on the set of $n$-colorings.

What is the difference between the Pólya Enumeration Theorem and Burnside's Lemma?

  1. The Pólya Enumeration Theorem is more general than Burnside's Lemma.

  2. Burnside's Lemma is more general than the Pólya Enumeration Theorem.

  3. The Pólya Enumeration Theorem and Burnside's Lemma are equivalent.

  4. The Pólya Enumeration Theorem and Burnside's Lemma are unrelated.

Show answer
Correct Option: A
Explanation:

The Pólya Enumeration Theorem is more general than Burnside's Lemma because it applies to any object with a symmetry group, while Burnside's Lemma only applies to objects that can be colored with a finite number of colors.

What are some applications of the Pólya Enumeration Theorem?

  1. Counting the number of ways to color a graph.

  2. Counting the number of ways to label a tree.

  3. Counting the number of ways to arrange a set of objects in a circle.

  4. All of the above.

Show answer
Correct Option: D
Explanation:

The Pólya Enumeration Theorem can be used to count the number of ways to color a graph, label a tree, or arrange a set of objects in a circle.

What are some applications of Burnside's Lemma?

  1. Counting the number of ways to color a Rubik's Cube.

  2. Counting the number of ways to tile a floor with a given set of tiles.

  3. Counting the number of ways to arrange a set of objects in a line.

  4. All of the above.

Show answer
Correct Option: D
Explanation:

Burnside's Lemma can be used to count the number of ways to color a Rubik's Cube, tile a floor with a given set of tiles, or arrange a set of objects in a line.

The Pólya Enumeration Theorem and Burnside's Lemma are important tools in what field of mathematics?

  1. Combinatorics

  2. Algebra

  3. Analysis

  4. Geometry

Show answer
Correct Option: A
Explanation:

The Pólya Enumeration Theorem and Burnside's Lemma are important tools in combinatorics, which is the branch of mathematics that deals with counting and arranging objects.

Who is George Pólya?

  1. A Hungarian mathematician who made significant contributions to combinatorics, number theory, and analysis.

  2. A French mathematician who made significant contributions to algebraic geometry and topology.

  3. A Russian mathematician who made significant contributions to functional analysis and operator theory.

  4. A British mathematician who made significant contributions to mathematical logic and set theory.

Show answer
Correct Option: A
Explanation:

George Pólya was a Hungarian mathematician who made significant contributions to combinatorics, number theory, and analysis. He is best known for the Pólya Enumeration Theorem, which is a fundamental result in combinatorics.

Who is William Burnside?

  1. A British mathematician who made significant contributions to group theory, number theory, and combinatorics.

  2. A French mathematician who made significant contributions to algebraic geometry and topology.

  3. A Russian mathematician who made significant contributions to functional analysis and operator theory.

  4. A German mathematician who made significant contributions to mathematical logic and set theory.

Show answer
Correct Option: A
Explanation:

William Burnside was a British mathematician who made significant contributions to group theory, number theory, and combinatorics. He is best known for Burnside's Lemma, which is a fundamental result in combinatorics.

In what year was the Pólya Enumeration Theorem first published?

  1. 1927

  2. 1937

  3. 1947

  4. 1957

Show answer
Correct Option: B
Explanation:

The Pólya Enumeration Theorem was first published in 1937.

In what year was Burnside's Lemma first published?

  1. 1901

  2. 1911

  3. 1921

  4. 1931

Show answer
Correct Option: A
Explanation:

Burnside's Lemma was first published in 1901.

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