I am trying to solve a question. I have 15 pens, where 5 are colored in Red, 5 in Green, and 5 in Blue. How many combinations can I order these pens so that there is no sequence of 5 pens with the same color? For instance, the combination RRRRG GGGBB BBRGB is okay because there is no sequence of pens where 5 colors are after each other. The combination RRRRR ... would need to be excluded because 5 red pens are in a row.
My initial strategy was to count the total number of combinations to build with these pens and then subtract the total number of combinations where at least 5 colors are in sequence.
The total number of combinations:
$$\frac{15!}{(5!*5!*5!)} = 756756$$
The issue I am having is how to create all the combinations where 5 colored pens are in order. I have tried to do something like this:
Group the sequence RRRRR into one symbol. Now, calculate every combination with this RRRRR symbol and the other symbols.
$$\frac{11!}{5!*5!}=2772$$
The problem here is that if I multiplied this with 3 to get the combinations from every color, there would be many duplicated combinations.
Any help on how I could solve this is much apprecited.