Permutations (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Permutations: core idea

Burnside peek.

**(Proof track.)** Give a complete justification. How many arrangements of $LEVEL$ have vowels adjacent?

Hint: Identify symmetries, repetitions, and whether blocks are needed. Answer: 12.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. How many circular seatings of eight people have A opposite B?
2. **(Proof track.)** Give a complete justification. How many permutations of $1$ through $7$ have $1,2,3$ in increasing order?
3. **(Proof track.)** Give a complete justification. How many six-digit even numbers use each of $0,1,2,3,4,5$ once?
4. **(Proof track.)** Give a complete justification. How many arrangements of $ABCDEFG$ have no two of $A,B,C$ adjacent?
5. **(Proof track.)** Give a complete justification. How many ways can five distinct people stand in line if A is not first or last?
6. **(Proof track.)** Give a complete justification. How many arrangements of $STATISTICS$ are there?
7. **(Proof track.)** Give a complete justification. How many permutations of $1,2,3,4$ have exactly two fixed points?
8. **(Proof track.)** Give a complete justification. How many ways can eight people be paired into four unordered pairs?
9. **(Proof track.)** Give a complete justification. How many strings from $A,A,B,B,C,C$ have no equal adjacent letters?