Could someone help me please with this problem? I don't even know how to start it.
For 1$\leq$k$\leq$n, find the number of permutations $\theta$ $\in$ $S_n$ that have exactly k fixed points.
Thank you in advance!
2026-02-23 22:22:31.1771885351
The number of permutations with k fixed points
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For the case $k=0$, this is the problem of derangements.
For $k > 1$, after first choosing $k$ members of $\{1,\ldots,n\}$ to be the fixed points, you then want a derangement of the remaining $n-k$.