Task 4. The twelve knights of the round table need to choose a two-member delegation to visit King Arthur. In how many ways can this be done so that the two-member delegation does not include knights who sit next to each other at the round table.
Solution. Let's assume that the round table is a circle, and the knights sitting at the round table are vertices of a 12-sided polygon. Each pair of knights sitting at the table, forming a two-member delegation according to the condition of the problem, defines one diagonal of the 12-sided polygon. Therefore, as many delegations can be formed as there are diagonals in the 12-sided polygon. The number of diagonals in a 12-sided polygon is equal to $\frac{12 \cdot 9}{2}=54$.

Thus, 54 two-member delegations can be formed in the manner specified in the problem.