Hello my question is quite simple i would think but i just cant seem to find an answer. I have a set of $\{1,2,3,4,5,6,7,8,9,10\}$ and i would like to calculate how many unique given sets of $6$ can i get from this set. In other words for the number $1$ i would end up with $[1,2,3,4,5,6] [1,3,4,5,6,7] [1,4,5,6,7,8] [1,5,6,7,8,9] [1,6,7,8,9,10]$ I would move down the line with the number $2$ to compare to unique sets of $6$ note: when moving to two I would no longer do this $[2,1,3,4,5,6]$ because it repeats my first case above. its there a formula to figure this sort of thing? Thanks in advance.
when I work this out on paper i end up with 15 sets here is how
for 1 [1,2,3,4,5,6] [1,3,4,5,6,7] [1,4,5,6,7,8] [1,5,6,7,8,9] [1,6,7,8,9,10] for 2 [2,3,4,5,6,7] [2,4,5,6,7,8] [2,5,6,7,8,9] [2,6,7,8,9,10] for 3 [3,4,5,6,7,8] [3,4,6,7,8,9] [3,5,6,7,8,9,10] for 4 [4,5,6,7,8,9] [4,6,7,8,9,10] for 5 [5,6,7,8,9,10]
after that i cant make any more groups of $6$ thus i end up with $15$ sets.