Question

$10$ persons are sitting in a row. In how many ways we can select three of them if adjacent persons are not selected$?$

Answer 1

Let $P_1,P_2,P_3,P_4,P_5,P_6,P_7,P_8,P_9,P_{10}$ be the persons sitting in this order.
If three are selected (non consecutive) tnen $7$ are left out.
Let $P,P,P,P,P,P,P$ be the left out & $q,q,q$ be the selected. The number of ways in which these $3$ q's can be placed into the $8$ positions between the P's (including extremes) is the number ways of required selection.
Thus number of ways${=}^8{\text{C}}_3=56.$