I've just seen The Story of God with Morgan Freeman and in the last episode a psychology professor says that drawing six cards from a 52 card deck has only a probability of one in 14 billion to be in that exact order.
I understand that the probability of getting these six cards can be calculated like this: $$\frac{1}{\dbinom{52}{6}}$$ but how do I calculate the probability of them being in exactly one order.
My thought is, that for the first spot there is a 1/6 probability, for the second a 1/5 probability and so on. The problem is I don't get how I should put this thought into an equation with the other equation.
Thanks for your help in advance.
There is one way to pick six particular cards in order.
How many ways can we select six of the $52$ cards in order?
We have $52$ options for the first card, $51$ options for the second card, $50$ options for the third card, $49$ options for the fourth card, $48$ options for the fifth card, and $47$ options for the sixth card. Hence, there are $$52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47 = \frac{52!}{46!} = \binom{52}{6}6!$$ ordered selections of six cards, so the desired probability is $$\frac{1}{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47} = \frac{1}{14~658~134~400}$$