Q.In how many ways 7 pictures can be hung from 5 picture nails on a wall ?

A permutation counts ordered arrangements — swapping two chosen objects' positions produces a different, distinct outcome. For $n$ distinct objects taken $r$ at a time, $^{n}P_{r} = n!/(n-r)!$. When some objects are indistinguishable from each other (repeated letters in a word, identical items in a batch), the count of distinct arrangements of all $n$ objects is reduced by dividing by the factorial of each repeated group's size: $n!/(p_1!\,p_2!\cdots)$. Arranging objects around a circle is a distinct sub-case, because rotating the whol …