Permutations and repeated symbols

The product rule multiplies numbers of choices at successive stages. n!=n(n-1)...1 and 0!=1. Arranging r distinct objects chosen from n without repetition gives nPr=n!/(n-r)!. If repeated symbols have multiplicities a,b,..., distinct arrangements of all n symbols number n!/(a!b!...). For n distinct people around an unlabelled circular table, with rotations equivalent and reflections distinct, the count is (n-1)!.