20 soldiers are standing in a row and their captain want to send 7 out of them for a mission in how many ways can captain select them such that at least one soldier and the soldier next to him is also selected?
The selection of 7 soldiers can be done in 20C7 ways....what is the next step?
The next step is to subtract the number of selections of $7$ where none of the selected soldiers are next to each other. To figure out that number, you can use stars and bars:
$$.*.*.*.*.*.*.*.*.*.*.*.*.*.$$
The stars are the $13$ soldiers that don't get selected, ans so the $7$ soldiers have to be a selection of the $14$ possible positions between them and at the ends. So, there are
$$14 \choose 7$$
selections where no two soldiers are next to each other.