Permutations with Identical Objects and with Restrictions

Permutations of $n$ Objects, Not All Distinct

If among $n$ objects one kind is repeated $p$ times, another $q$ times, another $s$ times (and so on), the number of distinct arrangements of all $n$ is

$\frac{n!}{p!\,q!\,s!\cdots}.$

The "Group Together" Technique

If certain objects must stay together, tie them into a single block, arrange the blocks, then multiply by the internal arrangements of the tied objects. For example, if $k$ specific objects among $n$ must be adjacent, treat them as one unit: arrange $(n-k+1)$ units, then multiply by $k!$ for the order inside the block.

The "Fix a Position" Technique

If an object must occupy (or avoid) a particular place, settle that place first, then arrange the rest freely in the remaining places. For a "never together" condition, it is often easiest to compute (total arrangements) $-$ (arrangements with them together). …