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.
$\endgroup$ 13 Answers
$\begingroup$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!
$\endgroup$ 5 $\begingroup$It exactly the number of ways to choose $6$ elements out of $10$, i,e. the binomial coefficient$$\binom{10}{6}=\frac{10!}{6!4!}$$
$\endgroup$ $\begingroup$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 $\frac{n!}{k!(n-k)!}$
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