Track / Stage 5 / 52 of 400 #652 of 1964

Problem 652

AIME late

Combinatorics Difficulty 5.1 Find the answer

Of the 31 students in a 4th grade class, 21 can swim, 24 can ride a bike, and 19 can ice skate. For a competition, students are needed who can

a) swim and ride a bike,

b) swim and ice skate,

c) ride a bike and ice skate,

d) swim and ride a bike and ice skate.

How many students from the class are available for a), b), c), and d) at a minimum, and how many at a maximum?

The source for this one didn't record the answer, so there is nothing to check what you type against. Work it on paper and mark yourself against the solution below.

Official solution

Maximal available

a) 21, as there are only 21 swimmers, b) 19, as there are only 19 skaters, c) 19, as there are only 19 skaters, d) 19, as there are only 19 skaters.

Minimal available
a) 14 because $24+21-31=14$ ,
b) 9 because $21+19-31=9$ ,
c) 12 because $24+19-31=12$ ,
d) 2 because $24+21+19-31-31=2$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.