Combinatorics problem

  • #1
BrownianMan
134
0
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

  • #2
BrownianMan
134
0
Anyone?
 
  • #3
Michael Redei
181
0
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)!}##.
 

Suggested for: Combinatorics problem

  • Last Post
Replies
9
Views
813
  • Last Post
Replies
13
Views
270
Replies
6
Views
861
  • Last Post
Replies
4
Views
150
  • Last Post
Replies
23
Views
359
  • Last Post
Replies
2
Views
753
  • Last Post
Replies
1
Views
2K
Replies
5
Views
123
Top