Combination and permutation of indistinguishable objects

To explain where your formula comes from and why it works, consider breaking it up into steps:

Step 1: choose where the reds go. There are $\binom{10}{2}$ ways of arranging the reds and not reds (ignoring the fact that the not reds are of multiple colors for the moment).

Step 2: of the spaces labeled for use by not-reds, choose which of those spaces will be occupied by blues: There are $\binom{10-2}{3}$ number of ways to do this.

Step 3: of the spaces labeled for use by not-reds and not-blues, choose which are occupied by greens: There are $\binom{10-2-3}{5}$ number of ways.

Thus, there are $\binom{10}{2}\cdot\binom{8}{3}\cdot\binom{5}{5} = \frac{10!~~~8!~~~5!}{2!8!3!5!5!0!} = \frac{10!}{2!3!5!}$ number of ways to accomplish this.

(remember that $0!=1$ by definition)

Determine if projection of 3D point onto plane is within a triangle

Calculate the Wu class from the Stiefel-Whitney class

Evaluation of $\lim\limits_{n\to\infty} (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) $

Prove that the field F is a vector space over itself.

Finding a trigonometric polynomial

Showing $\lim_{n \to \infty} m(E_n) = 0$, assuming $f > 0$ a.e. and $\lim_{n \to \infty} \int_{E_n}f \,dm =0$

Show that if $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$

ZF Extensionality axiom

maximal linear subspaces contained in the cone over the Clifford torus.

Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets? [duplicate]

Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers. [duplicate]