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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.

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    I'd be pretty certain there are other questions similar to this on the site that might help you.2012-12-02

3 Answers 3

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Yes, there is, it is called the binomial, written $\binom{n}{k}$, read $n$ choose $k$. The value is $\binom{n}{k}=\frac{n!}{k!(n-k)!}.$ So, in your case, you have $\binom{10}{6}=\frac{10!}{6!4!}=210.$ I hope you find this helpful!

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    THANK YOU! :-D i knew there was so$m$ething i was missing. Thank you! I truly appreciate your response.2012-12-02
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It exactly the number of ways to choose $6$ elements out of $10$, i,e. the binomial coefficient$\binom{10}{6}=\frac{10!}{6!4!}$

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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 $\frac{n!}{k!(n-k)!}$