A circular arrangement corresponds to a linear arrangement in which the ends of the line have been joined to form a circle.
We first count linear arrangements in which no two adjacent people are selected. Place $n - k$ blue balls in a row, leaving spaces between successive balls and at the ends of the row. There are $n - k - 1$ spaces between successive balls and two at the ends of the row, for a total of $n - k + 1$ spaces. Choose $k$ of these $n - k + 1$ spaces in which to place a green ball. Now number the balls from left to right. The numbers on the green balls, no two of which are adjacent, represent the positions of the selected people. Thus, the number of linear arrangements in which no two consecutive people are selected is $$\binom{n - k + 1}{k}$$
However, we have counted linear arrangements in which people at both ends of the row are selected. These arrangements are not permissible since when the ends of the row are joined to form a circle, the people seated at the ends of the row in a linear arrangement would be sitting in adjacent seats in a circular arrangement.
We count such linear arrangements. Place $n - k$ blue balls in a row, as before. Place a green ball at each end of the row. This leaves $n - k - 1$ spaces between successive blue balls and $k - 2$ green balls to place in those spaces. Therefore, there are $$\binom{n - k - 1}{k - 2}$$ linear arrangements in which people are seated at both ends of the row.
Consequently, the number of permissible circular arrangements in which no two of the $k$ selected people are adjacent is
$$\binom{n - k + 1}{k} - \binom{n - k - 1}{k - 2}$$
