Question

Let $N_{10} = \{1, 2, 3, ..., 10\}$. Consider a permutation $p$ of $N_{10}$. A point $k$ is fixed by $p$, if $p(k) = k$. A permutation with no fixed point is called derangement. Considering all permutations, what is the number of fixed points:

• A. 10
• B. 100
• C. 50
• D. 1

Answer 1

Answer: (D)

1

Answer 2

The question asks about the number of fixed points considering all permutations of $N_{10} = \{1, 2, ..., 10\}$. A fixed point of a permutation $p$ is an element $k \in N_{10}$ such that $p(k) = k$. The phrasing "the number of fixed points" is ambiguous.

The average number of fixed points per permutation is the total number of fixed points divided by the number of permutations: $\frac{10!}{10!} = 1$. The average number of fixed points in a random permutation of $n$ elements is always 1.