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.
Click here for Dr. Hari Kishan's explanation
Davit Banana gets extra credit for determining the number and probability for each possibilty of getting the correct helmet.
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.
Click here for Dr. Hari Kishan's explanation
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 |