2 More Difficult Questions, Any Help Is Appreciated
the first question is:
let a1, a2, a3, a4, a5 , be any distinct positive integers. Show that there exists at least one subset of three of these integers whose sum is divisible by 3.
the second quesiton is:
The simplex lock is a popular push-button combination lock found in aparment buildings, offices schools, etc... The metallic lock has five buttons that can be pushed individually or simultaneously with other buttons, and the order and number of pushes make up a unique code. Codes for the lock follow the protocol listed below
(i) A code has a sequence of zero or more pushes, each involving atleast on button.
(ii) Each button may be used at most once.
(iii) Each push may include any number of previously unpushed buttons.
(iv) When two or more buttons are pushed at the same time, order does not matter.
Suppose you have a five-buton simplex lock. How many possible codes exist?
plzzzzzz help on these crazy combinatorics questions