What is the number of ways to color n objects with 3 colors if every color must be used at least once?

Simple.
For the first object, we can use color 1
For the second object, we use color 2
For the third object, we use color 3.
After that, we can use any color we deem fit.
Apart from the 1st 3 objects, we have n ways of choosing the next color.
=> 1 * 1 * 1 * 3 * 3 * 3 ....
=> 1 * 3 ^ (n-3)
=> 3 ^(n-3)

n x 3
= 3n

You can find an explanation of this problem online. It's solution number 1.8.29. I like to see things solved in a logical, deductive matter then just spewing formula. The deductive reasoning shown in the source follows the thought process required for mathematical proof which is more insightful.