11.4. Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 boards so that any two adjacent boards are of different colors, and he uses all three colors of paint?
Answer: 1530 ways.

First, let's calculate the number of ways to paint the fence so that any two adjacent boards are painted in different colors. The first board can be painted with any of the three colors, the second with one of the two remaining colors. The third board can be painted with one of the two colors different from the color of the second board, and so on. Therefore, the number of ways is $3 \cdot 2^{9}=1536$. In this number, the ways to paint the fence using only two colors are also included. The number of such ways is 6 (the first board has 3 options, the second has 2, and the rest are uniquely determined). Therefore, $1536-6=1530$ ways.