Advanced Permutations and Cycle Structures Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) How many permutations of {1,2,…,n} map 1 to 3?

Explanation

Fixing 1→3 leaves (n−1) elements to permute.

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About This Quiz
Advanced Permutations And Cycle Structures Quiz - Quiz

Think you understand permutations at a deeper level? This quiz challenges you to apply graduate-level ideas such as cycle decompositions, fixed points, Stirling numbers of the first kind, and structural properties of bijections. You’ll evaluate how permutations behave when constraints are added, like fixing certain elements or limiting cycle lengths.... see moreAlong the way, you'll explore even/odd parity, derangements, and cycle-type counting. By the end, you’ll see how advanced combinatorics reveals hidden patterns and elegant structure in the symmetric group! see less

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2) Every permutation with no fixed points is a derangement.

Explanation

A derangement has zero fixed points.

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3) How many permutations of 7 elements begin with a 4 and end with a 2?

Explanation

Fix first & last → 5!.

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4) Number of permutations of n with exactly k cycles is:

Explanation

c(n,k) are Stirling numbers of first kind.

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5) Which are always true of permutations without repetition?

Explanation

Permutation = bijection using elements once.

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6) A permutation of only 2-cycles must have even order.

Explanation

Disjoint transpositions have order 2.

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7) How many permutations of 8 elements have 1 as a fixed point?

Explanation

Fix 1; permute 7 others.

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8) 6-letter arrangements from A–F, no repetition, starting with vowel?

Explanation

Two vowel choices then 4!.

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9) Permutations on 6 with cycle structure (3,2,1):

Explanation

6!/(3·2)=60.

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10) A permutation with a single cycle of length n is an n-cycle.

Explanation

Definition of n‑cycle.

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11) Which permutations of {1,2,3,4} are even?

Explanation

Even permutations have even number of transpositions.

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12) How many permutations of 9 send 1→2 and 2→1?

Explanation

Fix (1 2); permute 7 others.

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13) Disjoint cycles can be reordered without changing the permutation.

Explanation

Disjoint cycles commute.

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14) Permutations with exactly one 2‑cycle:

Explanation

Choose 2 for cycle then permute rest.

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15) Permutations of 7 with no cycles longer than 3:

Explanation

Sum of valid cycle‑type counts.

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How many permutations of {1,2,…,n} map 1 to 3?
Every permutation with no fixed points is a derangement.
How many permutations of 7 elements begin with a 4 and end with a 2?
Number of permutations of n with exactly k cycles is:
Which are always true of permutations without repetition?
A permutation of only 2-cycles must have even order.
How many permutations of 8 elements have 1 as a fixed point?
6-letter arrangements from A–F, no repetition, starting with vowel?
Permutations on 6 with cycle structure (3,2,1):
A permutation with a single cycle of length n is an n-cycle.
Which permutations of {1,2,3,4} are even?
How many permutations of 9 send 1→2 and 2→1?
Disjoint cycles can be reordered without changing the permutation.
Permutations with exactly one 2‑cycle:
Permutations of 7 with no cycles longer than 3:
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