Finding a combinatorial proof of this identity: $n!=\sum_{i=0}^n \binom{n}{n-i}D_i$

Question

Can someone prove this. Let $D_n$ be the number of derangements of $n$ objects. Find a combinatorial proof of the following identity: $$n!=\sum_{i=0}^n \binom{n}{n-i}D_i$$

Answer 1

You can divide all $n!$ permutations of $n$ objects onto $n+1$ disjoint classes of permutations with exactly $i$ dearrangements where $i=0,\ldots,n$. Amount of permutations with exactly $i$ dearrangements equals to the product of ways to choose objects that will be dearranged i.e. ${n \choose i}$ and amount of ways to dearrange them i.e. $D_i$. Thus $$ n! =\sum\limits_{i=0}^n {n \choose i} D_i =\sum\limits_{i=0}^n {n \choose n-i} D_i $$