(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If k people are seated in a random manner in a row containing n seats (n>k), what is the probability that the people will occupy k adjacent seats in the row?

I realize that there are n choose k sets of k seats to be occupied, and that there are n-k+1 sets of k adjacent seats. So the probability that I'm looking for is:

(n-k+1)/(n choose k)

What I don't understand is how the above probability simplifies to:

[(n-k+1)!k!]/n!

Can someone please explain? Thanks.

2. Relevant equations

(^{n}_{k}) = n choose k = n!/[(n-k)!k!]

3. The attempt at a solution

(n-k+1)/(n choose k) = [(n-k+1)(n-k)!k!]/n!

Not sure what to do from here.

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# Basic combinations problem

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