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Mathbound · Contest practice

Permutations worksheet

Mixed difficulty

Name

Date

1.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $11$ is unsafe. How many jump sequences reach pad $16$ without landing on pad $11$ ?
- A.445
- B.447
- C.446
- D.1597
- E.444
2.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $9$ is unsafe. How many jump sequences reach pad $22$ without landing on pad $9$ ?
- A.7924
- B.7923
- C.28657
- D.7921
- E.7922
3.
A bug walks on the edges of a cube. Starting at a vertex, how many walks of length $6$ return to the start?
- A.729
- B.183
- C.192
- D.186
- E.180
4.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $13$ is unsafe. How many jump sequences reach pad $20$ without landing on pad $13$ ?
- A.3029
- B.3031
- C.3030
- D.10946
- E.3028
5.
Ways to tile a $2\times10$ board with dominoes?
- A.89
- B.55
- C.144
- D.34
- E.233
6.
A frog starts at pad $0$ and jumps $+1$ or $+2$ each move. Pad $17$ is unsafe. How many jump sequences reach pad $18$ without landing on pad $17$ ?
- A.1598
- B.4181
- C.1596
- D.1597
- E.1599
7.
A bug walks on the edges of a cube. Starting at a vertex, how many walks of length $4$ return to the start?
- A.81
- B.21
- C.30
- D.24
- E.18
8.
How many involutions (bijections $f$ with $f\circ f=\mathrm{id}$ ) exist on a set of $7$ elements?
- A.255
- B.232
- C.128
- D.239
- E.225
9.
How many involutions (bijections $f$ with $f\circ f=\mathrm{id}$ ) exist on a set of $11$ elements?
- A.39266
- B.35696
- C.2048
- D.35707
- E.35685
10.
A bug walks on the edges of a unit cube. Starting at a vertex, how many walks of length $6$ along edges end at the **same** vertex?
- A.162
- B.183
- C.192
- D.216
- E.243

Answer key · worked solutions

1. A. 445
Fibonacci with a zeroed forbidden state. DP with $F(11)=0$ . $445$ .
2. E. 7922
Fibonacci with a zeroed forbidden state. DP with $F(9)=0$ . $7922$ .
3. B. 183
Cube graph adjacency powers / distance states. Length $6$ closed walks. $183$ .
4. A. 3029
Fibonacci with a zeroed forbidden state. DP with $F(13)=0$ . $3029$ .
5. A. 89
Fibonacci tiling. $a_{10}=89$ . $89$ .
6. D. 1597
Fibonacci with a zeroed forbidden state. DP with $F(17)=0$ . $1597$ .
7. B. 21
Cube graph adjacency powers / distance states. Length $4$ closed walks. $21$ .
8. B. 232
$I_n=I_{n-1}+(n-1)I_{n-2}$ — not a $4$ -element listing drill. Compute the recurrence up to $n=7$ . $232$ .
9. B. 35696
$I_n=I_{n-1}+(n-1)I_{n-2}$ — not a $4$ -element listing drill. Compute the recurrence up to $n=11$ . $35696$ .
10. B. 183
Cube graph adjacency; track distance from start ( $0,1,2,3$ ). Recurrence on distance types — not $3^6$ independent choices. $183$ .

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