Could you please solve this problem?

Question

15 persons are arranged in a row. Find the number of ways of selecting  6 persons so that no two persons sit next to each other.

I think we have to select 6 persons from either the group of 8 people sitting 1st, 3rd, 5th.....15th or the group of 7 people sitting 2nd, 4th...14th.

I tried to solve this way but I'm not getting the required answer. Please some one help me.

Answer 1

If you are selecting $6$ persons,

$9$ persons are left behind, and the selections must have been made from the $10$ gaps (see diagram)

$-P-P-P-P-P-P-P-P-P-$

thus $\binom{10}{6}$ ways

Answer 2

Hint: You can think of it as if you are choosing the gaps between the 6 persons. Let $x_1$ be the number of persons before the first chosen person, $x_2$ the number of persons between the first and second chosen persons, etc. Then you want to solve $x_1+x_2+\dots+x_7=9$, where $x_1,x_7\geq 0$ and $x_2,\dots,x_6\geq 1$.