Our math department has been having an interesting discussion around a relatively simple probability question. We think it makes sense to do it two different ways, but in so doing, we come up with two different answers.
Here's the question: What is the probability of drawing without replacement from a standard deck of 52 cards the following 5 card hand ...... the ace of spades, 2 tens and 2 face cards
Solution 1: 1/52 x 4/51 x 3/50 x 12/49 x 11/48
Solution 2: Using Combination theory ..... (1C1 x 4C2 x 12C2) divided by (52C5)
Can you help us understand which answer is correct and why the other is not?
Think of a simpler problem: What is the probability of drawing without replacement from a standard deck of 52 cards the following 2 card hand ...... the ace of spades and the 3 of hearts.
Solution 1 counts the probability of getting the 2 card hand in the precise order ace of spades followed by the 3 of hearts - 1/52 x 1/51. But you could get the hand in the opposite order - 3 of hearts followed by the ace of spades - with probability 1/52 x 1/51 also. When the cards are dealt to you, face down on the table, you can mix them around to your hearts content but when you pick them up you consider it the same hand, regardless of the order, ace-then-3 or 3-then-ace. The point is that a 'hand' has no order of the cards attached to it. That's why solution 2 would be correct: (1C1 x 1C1)/52C2; the 2 in the 52C2 takes care of the 2! = 2x1 possible orders of the two cards.
In your original problem a 5! appears in solution 2 to take care of the 120 possible orders that the cards could be dealt to you. Solution 1 is correct if you want the ace of spades then 2 tens and then 2 face cards in that order: it is in fact 1C1/52C1 x 4C2/51C2 x 12C2/50C2. Another way of saying this is that we're using conditional probabilities here. You have the probability of choosing the ace of spades, and given that you've done that, the probability of picking two tens is now 4C2/51C2, and given that you've done both of those, the probability of picking two facecards is now 12C2/49C2.