TITLE: Sheldon Ross vs My TA, what answer is wrong?
QUESTION [7 upvotes]: I have the solution of this problem,
1) The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is chosen randomly and the object of the game is to guess the chosen three.
In one version of the game, the selection is made and then each of the players is
randomly given three of the remaining cards. Let S, W, and R be, respectively, the
number of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let X denote the number of solutions that are possible after that player observes his or her three cards.
I need to find: (c) Find E[X].
Solution given by my TA:

Solution given by Sheldon Ross -
A first Course in Probability (Seven Edition).

My question, is of what answer is wrong and what answer is correct, and why?

REPLY [4 votes]: The TA is right, and Ross (or more likely whoever made the solution key for him) is wrong.
The deck has $21$ cards (6 suspects, 6 weapons, 9 rooms) from which the player is given three.  However, we know that one suspect, one weapon and one room have been removed from the deck.  Thus e.g. the probability that the player was given three room cards is not ${9 \choose 3}/{21 \choose 3}$,
as Ross's solution implies, but ${8 \choose 3}/{18 \choose 3}$.