A Ruby implementation of derangement (secret santa) algorithm
I'm a bit unsure about whether the question is about a game of Secret Santa or if it's actually about derangement, specifically. Derangement requires that no element can appear in its original position. But that's irrelevant for a game of Secret Santa, since there's no ordering to speak of.

Edit: As tokland rightly points out in the comments derangement is of course related to the game. I wasn't thinking it through.

For derangement, Rosetta Code has a Ruby implementation you might be able to use or learn from.

For a game of Secret Santa, though, the specs are pretty much as you summed up in your 3 rules (although #3 "Assignment must be random" is a bit difficult to define/verify.) So here, things can be simplified; you need not worry about derangement "by itself" as long as the rules are obeyed. For instance, you might simply do:

players = %w[Mohamad Carolina Sami Tania Ikram Jose].shuffle
players &lt;&lt; players.first # repeat the first player

assignments = players.each_cons(2).to_a


which will give you an array of santa/gift-receiver pairs such as:

[["Carolina", "Ikram"],
 ["Ikram", "Mohamad"],
 ["Mohamad", "Jose"],
 ["Jose", "Tania"],
 ["Tania", "Sami"],
 ["Sami", "Carolina"]]


Alternatively, you can get the "Santa targets" for several rounds by doing something like

players = %w[Mohamad Carolina Sami Tania Ikram Jose].shuffle
assignments = players.each_with_index.map do |santa, index|
  others = players.rotate(index+1)[0...-1]
  [santa, others]
end


Which will give you something like

[["Tania", ["Mohamad", "Sami", "Ikram", "Carolina", "Jose"]],
 ["Mohamad", ["Sami", "Ikram", "Carolina", "Jose", "Tania"]],
 ["Sami", ["Ikram", "Carolina", "Jose", "Tania", "Mohamad"]],
 ["Ikram", ["Carolina", "Jose", "Tania", "Mohamad", "Sami"]],
 ["Carolina", ["Jose", "Tania", "Mohamad", "Sami", "Ikram"]],
 ["Jose", ["Tania", "Mohamad", "Sami", "Ikram", "Carolina"]]]


There's a pattern of course (if you've given Ikram a gift, next you'll be giving Carolina a gift, then José, etc.). But if it's Secret Santa, the players shouldn't be able to figure that out anyway unless they cheat (and if they do, well, all bets are off).

You could of course use Array#combination or Array#permutation to achieve similar results. I highly encourage you to check out all of the built-in array methods, and those included from the Enumerable module. There's a lot of good stuff there.

As for a more OOP approach, I wouldn't make a class called "Derangement". Derangement is a method. It's an action, an operation, a means of achieving a certain result or state - not something that is itself stateful.
The simple solution, given your code, is to rename you class to "Game" or something along those lines.