Selection with condition The problem statement is : 16 people are occupying seats round a table. If ever person refuses to work with any of his neighbours, in how many ways can a committee of 6 can be made with those persons? How do I approach this type of problem?

Mainly to facilitate the explaning in the answer let's give the persons numbers $1,2,\dots,16$ clockwise.

Pick one person out - let's say number $1$ - to be the somehow virtual chairman of the committee.

Now we start looking for tuples $\left(n_{1},\dots,n_{6}\right)$ where the $n_{i}$ are positive integers and $n_{1}+\cdots+n_{6}=10$.

These $n_{i}$ stand for the cardinality of the gaps, i.e. consecutive persons that are not in the comittee.

If e.g. $n_{1}=2$ and $n_{2}=1$ then persons with number $4$ and number $6$ are the next in the committee.

Applying stars and bars) we find $\binom{4+5}{5}=126$ possibilities and each tuple $\left(n_{1},\dots,n_{6}\right)$ represents a committee having person $1$ as chairman.

There are $16$ choices for the chairman so we come to $16\times126=2016$ possibilities.

However, every possible committee has been counted $6$ times because there are $6$ choices for the chairman, so we end up with: $$\frac{2016}{6}=336$$ possibilities.