A box contains five set of balls while there are three balls in each set. Each set of balls has one colour which is different from every other set. What is the least number of balls that must be removed from the box in order to claim with certainly that a pair of balls of the same colour has been removed?

There are 5 distinct colours (one per set) with 3 balls each. In the worst case, you could remove one ball from each colour and have 5 balls all of different colours; no pair yet. Any further removal (the 6th ball) must match one of the five colours already taken, producing a pair. Thus the least number that guarantees a same-colour pair is 6, following the pigeonhole/worst-case argument as shown in the provided solution sketch .