Adding Gigantic Numbers!

There are 10 objects. You color $\alpha$ of them red, $\beta$ of them green, and $\gamma$ of them blue; there are $\frac{10!}{\alpha!\beta!\gamma!}$ ways to do this. Summing over all $\alpha, \beta, \gamma$ whose sum is 10 gives all possible colorings of the 10 objects with red, green, and blue. But we know that the number of such colorings is just $\boxed{3^{10}}$ , since each object can be colored independently in three ways.

Consider expanison of $(x+y+z)^{10}$ .

$\sum_{\alpha+\beta+\gamma=10}\frac{10!}{\alpha!\beta!\gamma!}$ is equal to o the sum of the coefficients in the expansion , which we get by putiing $x=y=z=1$

Therefore, $\displaystyle \sum_{\alpha+\beta+\gamma=10}\frac{10!}{\alpha!\beta!\gamma!}$ $=$ $\boxed{3^{10}}$ .

It is simply the sum of the coefficients of a polynomial distribution that is equal $3^{n}$ for an n times repetition of a test having 3 distinct final states each time. Compare this please with the binomial distribution, there we get $2^{n}$ .

The problem needs to be fixed: the solution only works if the numbers are non-negative (including $0$ ). If the numbers are positive, the answer is $55980$ .