ABC String

How many strings of A's, B's, and C's are there that satisfy the following conditions:

1. There are $n$ A's and $2n$ total of B's and C's.

2. Every two adjacent letters are different.

KBB2 Problem 2

The country of Sikinia uses gold, silver, and bronze coins as its currency. One gold coin is worth 334 silver coins, and one silver coin is worth 208 bronze coins. One day, Ali went to the store and bought an item priced at 3 gold coins. How many ways can Ali pay the item in, assuming that he has unlimited supply of gold, silver, and bronze coins?