Advanced Permutations and Cycle Structures Quiz

1) How many permutations of {1,2,…,n} map 1 to 3?

(n-1)!

N!

N!/2

(n-2)!

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

Explanation

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

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

True

False

A derangement has zero fixed points.

Explanation

A derangement has zero fixed points.

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

5!

7!

4!

3!

Fix first & last → 5!.

Explanation

Fix first & last → 5!.

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

(n¦k)

C(n,k)

S(n,k)

D_(n-k)

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

Explanation

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

5) Which are always true of permutations without repetition?

They are bijections

They preserve order

No element appears twice

All cycles same length

Permutation = bijection using elements once.

Explanation

Permutation = bijection using elements once.

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

True

False

Disjoint transpositions have order 2.

Explanation

Disjoint transpositions have order 2.

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

8!

7!

6!

2⋅7!

Fix 1; permute 7 others.

Explanation

Fix 1; permute 7 others.

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

2⋅4!

6!

5!

4!

Two vowel choices then 4!.

Explanation

Two vowel choices then 4!.

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

20

60

40

120

6!/(3·2)=60.

Explanation

6!/(3·2)=60.

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

True

False

Definition of n‑cycle.

Explanation

Definition of n‑cycle.

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

(1 2)

(1 2 3)

(1 3)(2 4)

Identity

Even permutations have even number of transpositions.

Explanation

Even permutations have even number of transpositions.

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

7!

8!

9!/2

6!

Fix (1 2); permute 7 others.

Explanation

Fix (1 2); permute 7 others.

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

True

False

Disjoint cycles commute.

Explanation

Disjoint cycles commute.

14) Permutations with exactly one 2‑cycle:

(n¦2)(n-2)!

N!/2

D_(n-2)

N!/(n-2)!

Choose 2 for cycle then permute rest.

Explanation

Choose 2 for cycle then permute rest.

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

301

408

720

103

Sum of valid cycle‑type counts.

Explanation

Sum of valid cycle‑type counts.

Advanced Permutations and Cycle Structures Quiz

1) How many permutations of {1,2,…,n} map 1 to 3?

2)

What first name or nickname would you like us to use?

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

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

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

5) Which are always true of permutations without repetition?

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

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

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

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

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

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

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

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

14) Permutations with exactly one 2‑cycle:

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