Number of Possibilities Using Set of Rules Let's say I have 5 variables (a,b,x,y,z). * Each variable $\in (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)$. * The length of each variable must be 3. So _a_ can be e.g. ${1,2,3}$ or ${2,3,4}$ or ${3,4,5}$, etc... * The numbers is each variable must be consecutive e.g. (5,6,7) or (11,12,13) My question is how do I calculate the possible variations of all 5 variables combind resulting in the above. Basically, what's the number of possible combinations of _n_ number of variables that satisfy the above rules. This is my very first question on the math site, so im definitely not using the right terminology or tags to describe the question

Given what was told in the comments and question, it follow that:

1. there are 13 possible consecutive sets for each variable$$\\{\\{1,2,3\\},\\{2,3,4\\},\\{3,4,5\\},\\{4,5,6\\},\\{5,6,7\\},\\{6,7,8\\},\\{7,8,9\\},\\{8,9,10\\},\\{9,10,11\\}\\{10,11,12\\}\\{11,12,13\\}\\{12,13,14\\}\\{13,14,15\\}\\}$$
2. with no overlap the first variable can be any one of these 13, the second can be 12, of them the third 11,fourth 10,fifth 9 and you are right we multiply these to get: 154440 . Then, we realize that since ordering of the variables, isn't considered important, then we can divide by 5!=120 to get 1287.
3. Of course, this doesn't consider, the no overlap in values. Once we do that, we have that there is only one way ( without ordering becoming important) that this can be done. $$\\{\\{1,2,3\\},\\{4,5,6\\},\\{7,8,9\\}\\{10,11,12\\},\\{13,14,15\\}\\}$$