Question

Twelve persons are arranged in a row. The number of ways of selecting four persons so that no two persons sitting next to each other are selected is:

Answer 1

Imagine selecting lined up numbered people using ↑s and not selecting with ∙s, then one such valid selection is

1↑2∙3∙4∙5∙6↑7∙8∙9∙10↑11∙12∙

This is the selection {1,6,10}

So you are really only counting arrangements of 3 ↑s and 9 ∙s such that there is at least 1 ∙ between each pair of ↑s. The arrangement above is

↑∙∙∙∙↑∙∙∙↑∙∙

So all we do is “bank” two ∙s (one for each of the two spaces between ↑s) and count arrangements of the remaining 7 ∙s with 3 ↑s, so that our example is the arrangement

↑∙∙∙↑∙∙↑∙∙

counting all such arrangements of 7 identical ∙s and 3 identical ↑s gives

[Math Processing Error]

You may recognise this as (103).

This approach generalises easily to selecting k people from n with gaps of size g.

Hope It Helped You Mate