$Number of Combinations Created from Restricting a Set$

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$Number of Combinations Created from Restricting a Set$

Here’s my question:

I have a set that looks like this:

S = {u, v , w , x , y , z}

And it has this restriction:

1 =< u < v < w < x < y < z <= 11

Essentially, of 6 values, all of them must be greater than or equal to 1, must be less than their successor, must be greater than their predecessor, and must be less than or equal to 11.

How many sets produced from this restriction?

Thank you in advance.

Set Theory

Darkened Flames

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Answer 1

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Suppose you choose $6$ different integers between 1 and 11. Then you can sort them in the increasing order
\[ 1 \leq u < v < w < x < y < z \leq11,\]
where $u$ is the smallest number you have chosen, $v$ is the second smallest and so on. Thus the number of sets you are intersted in is equal to the number of ways you can choose $6$ different object from 11 different objects which is
\[{11 \choose 6}=\frac{11!}{(11-6)! 6!}=462.\]
In fact the restriction you have is equivalent to the restriction that the chosen numbers are different. If the 6 numbers are different, then they can be sorted in an increasing order in a unique way and hence satisfy the restriction.

Erdos

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Darkened Flames

0

Great answer and really intuitive. If you were interested, this is actually a question in disguise that is analyzing the number of scales that can be formed with 7 notes. Thank you so much :)
Erdos

0

Interesting! I am glad I was able to help.

$Number of Combinations Created from Restricting a Set$

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