Combinatorics Problem - Picking Non-consecutive Numbers

Описание: John shows two different methods for counting the number of ways of picking 3 numbers [none of which are consecutive] from {1, ... , 10}. https://www.swartwoodonline.com/cours...
www.swartwoodONLINE.com
www.swartwoodprep.com

One solution involves translating the problem to a_1 less than (a_2) - 1 less than (a_3) -2. Another solution involves visualizing 7 "rejected numbers" and placing the "accepted numbers" between them [relying on the fixed ordering 1-10 from left to right.