Probability of related events

Question

I was presented with the following problem:

Choosing three cards from a deck of 52, what is the probability of getting 2 Kings and a Queen?

I had two thought processes that both led to different results:

1. (4 * 3 * 4) / 3! [3! to eliminate a set order caused by the multiplication] all over (52 3)

2. (4 2) (4 1) / 2! [2! to, again eliminate order] all over (52 3)

Checking with other online sources however, (4 2) (4 1) [without /2] is correct meaning that neither of mine are.

These answers are based on my observation that when multiplication occurs, such as in the problem of "choose a first place, second place, and third place," the answer is always: n(n-1)(n-2) this indicates a fixed order, so to remove this order, we divide by the (number of terms!), so can anyone explain why any of these is wrong, and why my observation does / does not make sense?

Answer 1

Let K=drawing a king

and Q=drawing a queen

4 kings and 4 queens in a 52 card deck

Find Probability of drawing KKQ, KQK and QKK. Then add them together

P(KKQ) = 4/52 x 3/51 x 4/50

P(KQK) = 4/52 x 4/51 x 3/50

P(QKK) = 4/52 x 4/51 x 3/50

add those 3 probabilities 4/52 = 1/13

they sum to (1/13)(3/51 x 4/50 + 4/51x3/50 + 4/51x3/50)

= (1/13)(3/51 x 2/25 + 2(4/51x3/50))

= a very small probability. Don't bet on it, without major odds in your favor for the payoff