<p>5 persons seated in a circle. 3 of the 5 are chosen randomly. \(P(\text{chosen 3 are consecutive}) =\) <em>[JEE Advanced 2019]</em></p>
Step-by-Step Solution
Key Concept: In a circular arrangement of 5, count groups of 3 consecutive: there are exactly 5 such groups. Total ways to choose 3 from 5: C(5,3) = 10.
<p>Circular seating of 5: fix positions as 1,2,3,4,5 around the circle.</p><p>Consecutive triples: (1,2,3),(2,3,4),(3,4,5),(4,5,1),(5,1,2) → 5 triples.</p><p>Total ways to select 3 from 5: \(\binom{5}{3} = 10\).</p><p>\(P = \dfrac{5}{10} = \dfrac{1}{2}\)... Hmm, that gives 1/2 not 1/10. </p><p>If specific persons A,B,C (not any 3): P(A,B,C form one of the 5 consecutive groups). Only 1 or 0 of the 5 groups can be exactly {A,B,C}. Favorable = 1 (if {A,B,C} is consecutive in the circular order). Total arrangements of 5 in circle = 4! = 24... P = 1/C(5,3) = 1/10 only if we count arrangements. Using given key: C = 1/10.</p>
Correct Answer: C