Permutations & Restricted Arrangements

$n$ distinct objects: $n!$ arrangements. With repeats — $n$ objects where one letter appears $p$ times, another $q$ times, etc.: $\dfrac{n!}{p!\,q!\cdots}$. (E.g. MATHEMATICS: 11 letters with M, A, T each twice ⇒ $\dfrac{11!}{2!\,2!\,2!}$.)

• Together (block): treat the $k$ items as one unit ⇒ $(n-k+1)!$ for the units $\times\,k!$ inside.
• Apart / alternating: arrange the unrestricted items, then choose gaps for the rest.
• Fixed positions: fill the restricted positions first (e.g. vowels into the even slots), then fill the remaining positions.
• Dictionary (lexicographic) rank of a word: sort the letters; for each position in turn, count the letters still unused that are alphabetically EARLIER than the letter actually there, and multiply by the factorial of the remaining positions; add these up and add 1 for the word itself. Rank of RANK: letters A,K,N,R — before R come A,K,N ($3\cdot3!=18$); at A none; at N comes K ($1\cdot1!=1$); at K none; rank $=18+0+1+0+1=20$. With repeated letters divide each block count by the repeat factorials.