A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white…
A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. Consider the following statements:
1. The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is 175.
2. The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is 167.
Which of the above statements is/are correct?
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Solution
Statement 1: To avoid getting a full group of any colour, draw one less than each group: $13 + 19 + 25 + 27 + 37 + 53 = 174$. The next ball (175th) completes some full group. So n = 175 is correct. Statement 2: To avoid having a ball of every colour, miss one colour entirely; the worst case is missing the smallest group (14 black) — draw all others $20+26+28+38+54 = 166$. The 167th ball must be black, giving all colours. So m = 167 is correct. Both statements are correct.