Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a
drawer. Without looking, he randomly pulls shirts from his drawer one at
a time. He would like a set of shirts that includes either 3 of the same
colour or 3 of different colours. What is the minimum number of shirts
that Zebadiah has to pull out to guarantee that he has such a
set?
Problem 536
Pick one
Official solutions — 2
Solution 1
Zebadiah must remove at least 3 shirts.
If he removes 3 shirts, he might remove 2 red shirts and 1 blue
shirt.
If he removes 4 shirts, he might remove 2 red shirts and 2 blue
shirts.
Therefore, if he removes fewer than 5 shirts, it is not guaranteed that
he removes either 3 of the same colour or 3 of different colours.
Suppose that he removes 5 shirts.
If 3 are of the same colour, the requirements are satisfied.
If no 3 of the 5 shirts are of the same colour, then at most 2 are of
each colour. This means that he must remove shirts of 3 colours, since
if he only removed shirts of 2 colours, he would remove at most shirts.
In other words, if he removes 5 shirts, it is guaranteed that there are
either 3 of the same colours or shirts of all 3 colours.
Thus, the minimum number is 5.
Solution 2
Zebadiah must remove at least 3 shirts.
If he removes 3 shirts, he might remove 2 red shirts and 1 blue
shirt.
If he removes 4 shirts, he might remove 2 red shirts and 2 blue
shirts.
Therefore, if he removes fewer than 5 shirts, it is not guaranteed that
he removes either 3 of the same colour or 3 of different colours.
Suppose that he removes 5 shirts. If 3 are of the same colour, the
requirements are satisfied.
If no 3 of the 5 shirts are of the same colour, then at most 2 are of
each colour (for example, 2 red, 2 blue and 1 green). This means that he
must remove shirts of 3 colours, since if he only removed shirts of 2
colours, he would remove at most $2 + 2 =
4$ shirts.
In other words, if he removes 5 shirts, it is guaranteed that there are
either 3 of the same colour or shirts of all 3 colours.
Thus, the minimum number is 5.