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[Course Description]
This course prepares students to efficiently and effectively create and maintain video tutorials. When new video editing techniques are introduced, instructors will have the skills to evaluate these. Choose and use audio editing tools efficiently to produce both lower and higher quality content, such as. *Choose and use video editing tools to create video tutorials that are optimized for the Web as well as the Web-enabled devices or applications. *Create tutorials that are optimized for the Web and the Web-enabled devices or applications. *Design for both online and offline presentations, such as online webinars and offline content on DVDs, and.
[Course Description]
This course prepares students to efficiently and effectively create and maintain video tutorials. When new video editing techniques are introduced, instructors will have the skills to evaluate these. Choose and use audio editing tools efficiently to produce both lower and higher quality content, such as. *Choose and use video editing tools to create video tutorials that are optimized for the Web as well as the Web-enabled devices or applications. *Create tutorials that are optimized for the Web and the Web-enabled devices or applications. *Design for both online and offline presentations, such as online webinars and offline content on DVDs, and.Q:

Enumerate all subsets of $X$ having size $k$

let $A$ be a set with $n$ elements, and let $k \geq 1$ be an integer.
I am trying to prove that there are $k! \, n^k$ subsets of $A$ having size $k$.
By hand, it’s easy to show that in fact the number is
$$
(n-k+1)^{k-1} \cdot k!
$$
I don’t understand where the $(n-k+1)^{k-1}$ comes from. Any ideas?

A:

You are dividing the set of all possible $k$-subsets of the set $A$ into subsets of cardinality $k$ by dividing through by $n!$. A couple of the simpler cases of this:

For $n=2$, the set of all $1$-element subsets is just one element, the empty set. There are $2$ subsets of size $1$, so a total of $2$ ways to pick an arbitrary one-element subset of $A$.
For $n=3$, the set

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