Question

suppose we want to choose 5 colors, without replacement, from 8 distant colors.

Answer 1

When the order is relevant

$8P5=\frac{8!}{(8-5)!}=\frac{8!}{3!}=8\cdot7\cdot6\cdot5\cdot4=6720$

When the order it is no relevant:

We get that this problem is equivalent to say "how many ways are to choose 5 from a group of 8" so the answer is

$8C5=\frac{8!}{(8-5)!\cdot5!}=\frac{8!}{3!\cdot5!}=\frac{40320}{6\cdot120}=56$

so there are 56 ways to choose 5 colors from 8 without replacement