Q.If the letters of the word FUNNY are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, find the rank of the word FUNNY.

Real arrangement problems rarely ask for a plain $n!$ or $^nP_r$ — they add a constraint (some objects repeat, or must/must-not sit together, or are confined to particular positions). Four standard techniques handle these.

1. Repeated objects (Theorem 4.4). If among $n$ objects, $p_1$ are alike of one kind, $p_2$ alike of a second kind, ..., $p_k$ alike of a $k$th kind (rest all different), the number of distinct arrangements is

since permuting the identical copies of any one kind among themselves produces no visibly new arrangement — so the naive $n!$ must be divided by $p_i!$ for every repeated kind.

2. Objects that must stay together — the string/block method. Bundle the $m$ objects that must stay adjacent into a single unit. This leaves $(n-m+1)$ units to permute ($(n-m+1)!$ ways); the $m$ bundled objects also permute among themselves inside the bundle ($m!$ ways). Total: $m!\times(n-m+1)!$.

3. Objects that must never be adjacent — the gap method. Arrange the $m=n-k$ unrestricted objects first ($m!$ ways); this creates $m+1$ gaps (including both ends). Place the $k$ restricted objects into these gaps, at most one per gap, in $^{m+1}P_k$ ways. Total: $m!\times{}^{m+1}P_k$. (The complementary count — total unrestricted permutations minus the "always together" count — is often a faster route to the never-together answer.) …