If the goal was to find the number of different words that could be produced from scrambling a word that has different letters then that number would be n!,with n being the number of letters of that word.There is a way of finding that number in words where letters can be repeated.That way has to do with multiplying combinations using the "rule of product",or,in other words,dividing the permutations by a product of factorials.
Can anyone explain how that (the second) concept works,step-by-step?
Appreciate it!
Consider a word of length $n$, having $k$ distinct letters such that the first letter is repeated $r_1$ times, the second letter is repeated $r_2$ times, ....and the $k^{th}$ letter is repeated $r_k$ times.
First, we permute all $n$ letters. Then, notice that in any particular arrangement, shifting any of the repated letters among themselves wouldn’t make a difference. In other words, corresponding to each of the $n!$ arrangements, there exist $(r_1) !$ identical arrangements due the the presence of the first letter, and so on for every letter. Thus, the total number of words will be $$\frac{n!}{\prod_{i=1}^k (r_i)!}$$