The number of different
possible arrangements of 52 cards is 52! (= 52 x 51 x 50 x
… x 3 x 2 x 1).
That’s because there are 52 ways you can choose the first
card,
51 ways to choose the second card (once the first is chosen) 50 ways to
choose the third once the first two are chosen, etc).
Now, let’s
suppose the chosen
cards are Queen (Q) and King (K) (the following argument works
irrespective of what the values are). The first thing we need
to
think about is how many ways we can arrange a Q and K consecutively.
There are 4 four
Q’s (Qc, Qs, Qh, Qd) and four K’s (Kc, Ks, Ks, Kd).
This means there are 8
ways involving the Queen of Spades (Qs):
QsKs
KsQs QsKc
KcQs
QsKd KdQs
QsKh
KhQs
Similarly there are 8
ways involving
each of the other three Qs. Hence the total number of ways to arrange a
Q and K consecutively in two fixed positions is 32. But there are 51
such fixed positions, namely positions (1,2), (2,3), (3,4) …
(51, 52) in the deck. And for EACH of these fixed positions there are
50! Ways in which the remaining 50 cards can be arranged.
That means there are 32
x 51 x 50! possible arrangements of the cards in which a Q
and K appear consecutively.
So the probability of a
Q and K
appearing consecutively is simply this number divided by the total
number of possible arrangements, i.e.
