1987 IMO Problems/Problem 1
Problem
Let be the number of permutations of the set , which have exactly fixed points. Prove that
.
(Remark: A permutation of a set is a one-to-one mapping of onto itself. An element in is called a fixed point of the permutation if .)
Solution
The sum in question simply counts the total number of fixed points in all permutations of the set. But for any element of the set, there are permutations which have as a fixed point. Therefore
,
as desired.
Solution 2
The probability of any number where being a fixed point is . Thus, the expected value of the number of fixed points is .
The expected value is also .
Thus, or
1987 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
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