Answer
$54,264$
Work Step by Step
If we want to choose $k$ elements out of $n$ disregarding the order (because the order of colors does not matter), not allowing repetition, we can do this with combinations. The combination formula is:
$C(n,k)=\frac{n!}{(n-k)!k!}$ ways
Here, the order does not matter and hence we use combinations:
$\require{cancel}C(21,6)=\frac{21!}{(21-6)!6!}=\frac{21\cdot20\cdot19\cdot18\cdot17\cdot16\cdot15!}{15!6!}=\frac{21\cdot20\cdot19\cdot18\cdot17\cdot16\cdot\cancel{15!}}{\cancel{15!}6!}=\frac{21\cdot20\cdot19\cdot18\cdot17\cdot16}{6!}=54264$
