Probability

Concept:Probability = (favorable arrangements) / (total arrangements).Explanation:Total people = $8$ (X, Y, and 6 others).Total arrangements = $8!$.We need exactly 3 persons between X and Y, so their positions differ by $4$.Possible ordered position pairs: $(1,5), (2,6), (3,7), (4,8)$ and the reverses — $8$ ordered pairs.For each pair, place X and Y in those positions; arrange the remaining 6 people in $6!$ ways.Favorable arrangements = $8 \times 6!$.Probability = $\dfrac{8 \times 6!}{8!} = \dfrac{8 \times 6!}{8 \times 7 \times 6!} = \dfrac{1}{7}$.Answer:$\frac{1}{7}$