Permutations (Hard)

Late AMC 10/12 through early AIME — dense computation.

Permutations: core idea

Cycles/derange intro.

How many arrangements of $BANANA$ are there?

Hint: Decide whether order matters, then account for blocks and repeated objects. Answer: 60.

1. Six people sit around a circle. How many seatings have A and B adjacent?
2. How many five-digit numbers use distinct digits from $0,1,2,3,4,5$ ?
3. How many permutations of $1,\ldots,6$ have $1$ before $2$ and $3$ before $4$ ?
4. How many arrangements of eight books keep three specified books consecutive?
5. How many four-letter strings from $A$ -- $E$ have no repetition and start or end in $A$ ?
6. How many arrangements of $MISSISSIPPI$ are there?
7. Seven runners finish. In how many orders is C ahead of both D and E?
8. How many permutations of $1,2,3,4,5$ avoid every increasing consecutive adjacency?
9. Four married couples sit in a row, each couple together. How many arrangements?