we need to find the number of k-permutations of n element P(n, k) = n!/(n-k)! where n! is defined by n! = 1*2*..*(n-1)*n

in our case n = 10 and k = 5 thus

P(10, 5) = 10!/(10 - 5)!

P(10, 5) = 10!/5!

P(10, 5) = 6*7*8*9*10

P(10, 5) = 30240

there are 30240 ways of selecting the colors.

Note:

Permutation is usually understood to be a sequence containing each element from a finite set once, and only once. The concept of sequence is distinct from that of a set, in that the elements of a sequence appear in some order: the sequence has a first element (unless it is empty), a second element (unless its length is less than 2), and so on.