Thirty different integers are chosen at random from the integers 1-50 inclusive. Which of the following statements must be true?

I. At least on of the integers chosen is odd

II. At least one of the integers chosen is a multiple of 5

III. At least on one of the integers choses is less than 15

A) I
B) II
C) I and II
D) I and III
E) I, II and III

There are only 25 even integers in that range, so I is true.

There are 10 multiples of 5 in that range, so it is possible to pick 30 numbers that are not multiples of 5, so II doesn’t have to be true.

There are 14 integers less than 15, so III also doesn’t have to be true. One could pick 30 integers that are all greater than 15 and less than 50.

The answer is A.