Proof: Let s=(1, 2,.., n andx be the set of all permutations of s. Then x=n For j=1, 2,e., n, let pi be the property that in a permutation,j is in its natural position. Thus the permutation i1,i2, ...,in of s has property p provided i=j. A permutation of s is a derangement if and only if it has none of the properties p1p2…pn Let ai denote the set of permutations of s with property p; (j=1, 2,., n)▪ Proof: Let S={1,2,…,n} and X be the set of all permutations of S. Then |X|=n!. ▪ For j=1,2,…,n, let pj be the property that in a permutation, j is in its natural position. Thus the permutation i1 ,i2 ,…,in of S has property pj provided ij=j. A permutation of S is a derangement if and only if it has none of the properties p1 ,p2 ,…,pn . ▪ Let Aj denote the set of permutations of S with property pj ( j=1,2,…,n)