[Math] Find the probability that k people sits adjacent in a row of n seats

Question

I was solving a problem in probability (basic level). The problem is from a text book (Morris H.DeGroot). The problem is as,

if $k$ people are seated random in a row containing $n$ seats ( $n > k$ ). What is the probability that people will occupy $k$ adjacent seats in the row.

My approach of the solution is: Sample space is clearly ${n \choose k}$.
Now to get the event's possible outcomes I assume that 'k' adjacent seats are as 1 unit (as all the 'k' should be adjacent) then subtracting 'k' from 'n' and adding 1 unit to 'n' gives $(n-k)+1$. Now from this we just choose '1'. so it will be ${n-k+1 \choose 1}$
.

So the answer is $(n-k+1) \times \frac{1}{{n \choose k}}$. I see that the answer is correct from the text book answers. Is my approach correct? If not please let me know what should be the approach to solve this problem.

Thanks

Answer 1

Your answer is correct, and with a very high probability this is the most simple approach you could find :)