Difference between revisions of "2005 PMWC Problems/Problem I11"
m (sol) |
|||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| − | There are 4 men: A, B, C and D. Each has a son. The four sons are asked to enter a dark room. Then A, B, C and D enter the dark room, and each of them walks out with just one child. If none of them comes out with his own son, in how many ways can this happen? | + | There are <math>4</math> men: <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math>. Each has a son. The four sons are asked to enter a dark room. Then <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> enter the dark room, and each of them walks out with just one child. If none of them comes out with his own son, in how many ways can this happen? |
== Solution == | == Solution == | ||
| Line 9: | Line 9: | ||
When <math>n = 4</math>, we get: | When <math>n = 4</math>, we get: | ||
| − | <cmath>4! \left(\frac{1}{1} - \frac{1}{1} + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\right) = 9</cmath> | + | <cmath>4! \left(\frac{1}{1} - \frac{1}{1} + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\right) = \boxed{\textbf{9}}</cmath> |
== See also == | == See also == | ||
Latest revision as of 19:36, 2 March 2025
Problem
There are 
men: 
, 
, 
and 
. Each has a son. The four sons are asked to enter a dark room. Then , , and enter the dark room, and each of them walks out with just one child. If none of them comes out with his own son, in how many ways can this happen?
Solution
This is just a derangement problem (sending the set 
to another set such that none of the original elements are in the same place). The formula is:
![\[n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}\]](http://latex.artofproblemsolving.com/3/c/5/3c51ce639c3d03bba1ce6959ef27441312671127.png)
When 
, we get:
![\[4! \left(\frac{1}{1} - \frac{1}{1} + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}\right) = \boxed{\textbf{9}}\]](http://latex.artofproblemsolving.com/5/b/c/5bc4a744851b6b6e4898d087a4dc8bc6ea0e8b47.png)
See also
| 2005 PMWC (Problems) | ||
| Preceded by Problem I10 |
Followed by Problem I12 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
