On a table there are 10 different objects that can be colored using 5 different colors
a) How many ways can the 10 objects be colored? $10 \choose 5$
b) How many ways can the 10 objects be colored using all different colors? $5! \cdot 5!$ because I have to order 5 colors in 10 places, but when 5 places are "colored" I have 5 more place to color.
c) How many ways can the 10 objects be colored in such a way that each color is used at most 3 times? any idea?
Are my first two solutions right? what's about the third question? maybe $ 10 \choose 5 $ – $10 \choose 2$?
Best Answer
Those are not correct answers. Your solution before the edit was considering identical objects and used stars and bars. The edited solution does not answer either. The question says different objects. So for the first question, please note there are $5$ color choices for each object and hence the answer should simply be $ \ \displaystyle 5^{10}$ ways.
The second question seeks that all colors must be used (In contrast, in the first question, it is possible that all objects get colored using only one color as an example). So one approach is to use Principle of Inclusion Exclusion.
The answer to the second question should be,
$\displaystyle 5^{10} - {5 \choose 1} \cdot 4^{10} + {5 \choose 2} \cdot 3^{10} - {5 \choose 3} \cdot 2^{10} + {5 \choose 4}$
Alternatively use Stirling Number of the second kind (wiki) which gives you number of ways to arrange $10$ different objects in $5$ identical heaps. We then multiply by $5!$ to assign $5$ different colors to those heaps.
$\displaystyle 5! \cdot {10 \brace 5}$
Here is a hint for the third: Think if each color is used at most $3$ times, how many minimum colors must be used to color all $10$ objects and how many cases do you get? That is one approach.