January 2023
Problem of the Month

Bicycle Helmets
from the 6th Polya Mathematics Competition 1998

Six students traveled to a mathematics competition by bicycle.

On leaving, they distractedly picked up their six helmets at random (each choice of helmet is equally likely).

What is the probability that exactly three of the students picked up the correct helmet?

Solution to the Problem:

The probability is 1/18.

The students can pick up their helmets in 6!, or 720 possible ways.

Choose three of the 6 students to get the correct helmet, which can be done in
ways.

Then none of the remaining three people get the correct helmet, which can be done in two ways (bca or cab), where abc designates that everyone receives the correct helmet.

So the total is 20 x 2 = 40 out of 720, or 1/18.

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Click here for Dr. Hari Kishan's explanation

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Davit Banana gets extra credit for determining the number and probability for each possibilty of getting the correct helmet.

exactly no	number	probability
0	265	37% (53/144)
1	264	37% (11/30)
2	135	19% (3/16)
3	40	6% or (1/18)
4	15	2% (1/48)
5	0	0%
6	1	0% (1/720)
total	720	100%

Correctly solved by:

1. Colin Bowey	Beechworth, Victoria, Australia
2. Davit Banana	Istanbul, Turkey
3. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
4. Kelly Stubblefield	Mobile, Alabama

Send any comments or questions to: David Pleacher