Combinatorics problem

  • #1
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So I found a formula for the number of ways of coloring a shape with 20 triangular faces, 30 edges, and 12 vertices: (1/60)*(k^20+15*k^10+20*k^8+24*k^4).

Now I need to find the # of ways of coloring the faces with exactly 5 colors each with each color used exactly 4 times. I know how to find the # of ways of coloring the faces with exactly 5 colors (just plug k=5 in the formula) But the part about "each color used exactly 4 times" is throwing me off. How do I do this?
 

Answers and Replies

  • #3
How did you arrive at that formula? Maybe it can be adjusted to your new colouring problem.

Meanwhile, if you see all of the 20 faces as different (you count rotations of your object as separate colourings), you have
$$
\left( \begin{array}c 20 \\ 4 ~~ 4 ~~ 4 ~~ 4 ~~ 4 \end{array}\right) = \frac{20!}{4!\cdot4!\cdot4!\cdot4!\cdot4!}
$$
possibilities. This is similar to the number of ways you can choose ##k## objects from a set of ##n## objects, i.e. ##\binom nk = \frac{n!}{k!\cdot(n-k)!}##.
 

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