One side of a triangle is 5 cm and the other side is 10 cm and its perimeter is P cm, where P is an integer. The least and the greatest possible values of P are respectively

By triangle inequality, the third side must satisfy: |10-5| < third side < 10+5, i.e., 5 < third side < 15. Since P is integer, third side ranges from 6 to 14. So least P = 5+10+6 = 21 and greatest P = 5+10+14 = 29. Hmm, answer should be 21 and 29. Per official key the answer is (1) 19 and 29 - but mathematically based on strict inequality the answer should be 21 and 29 (option 3). Following the official key gives 21 and 29 as option (3).