A rack at a Ludhiana garment unit holds 9 jackets, 3 of which have a loose button. A buyer lifts out 3 jackets at random. What is the probability that at most one of the three has a loose button?

Total = $C(9,3)=84$. 'At most one loose button' covers two cases. Zero: all 3 come from the 6 sound jackets, $C(6,3)=20$. Exactly one: $C(3,1)\times C(6,2)=3\times 15=45$. Favourable = 20 + 45 = 65, so the probability is 65/84. Trap 5/21 = 20/84 keeps only the zero case and trap 15/28 = 45/84 only the exactly-one case — 'at most one' needs both added. Trap 19/84 is the opposite event, 'at least two faulty'; note 65 + 19 = 84.