Let n be a positive integer. Find the number of permutations of such that no number remains in its original place.
Solution: To do this, first we are going to count the number of permutations where at least one number remains in its place, according to the inclusion-exclusion principle, we must first add the permutations where a given number is fixed, then subtract the permutations where 2 given numbers are fixed and so on.
To find a permutation that fixes given elements, we only have to arrange the rest, which can be done in ways. However, if we do this for every choice of elements, we are counting permutations. Since in total there are permutations we get as our result:
I'm a bit confused about a point of the solution, when we fix elements and rearrange the other , some of the remaining elements will stay fixed in their position right? so why does this work?