How to calcualate how many unique set of 6 can i have in a given set. 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.          

Binomial coefficients count the number of distinct subsets of $k$ elements from a set containing $n$ elements.  The notation for this is $\binom{n}{k}$ which is equal to $\frac{n!}{k!(n-k)!}$
