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Problem 499

Combinatorics Difficulty 2.4 Multiple choice CEMC Cayley · Canada · 2023

Each of five doors is randomly either open or closed. What is the
probability that exactly two of the five doors are open?

Pick one

Official solution

Since each door can be open or closed, there are 2 possible
states for each door.

Since there are 5 doors, there are $2^5 = 32$ combinations of states for the 5 doors.

If the doors are labelled P, Q, R, S, T, the pairs of doors that can be
opened are PQ, PR, PS, PT, QR, QS, QT, RS, RT, ST. There are 10 such
pairs.

Therefore, if one of the 32 combinations of states is chosen at random,
the probability that exactly two doors are open is $\frac{10}{32}$ which is equivalent to
$\frac{5}{16}$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.