1. You Can’t Beat the Odds 3
We can easily ﬁnd similar examples from our common experience:
• For n = 52 and k = 5 we are asking about the number of
possible hands in poker.
• If at the close of a committee meeting each of the fourteen
members shakes hands in parting with each of the others,
how many handshakes are involved? Here we have the case
n = 14, k = 2.
And now back to the general problem. The formula that we are
seeking is a fraction with numerator n · (n − 1) ··· (n − k + 1) and
denominator 1 · 2 ··· k. The numerator may look a bit frightening to
the uninitiated, but it is simply the product of the k whole numbers
counting down by 1 from n. (Those interested in learning about where
this formula comes from will ﬁnd an introduction in Chapter 29.)
Here are a few additional examples:
• For the lottery problem, we must divide 49 ·48·47·46·45·44
by 1 · 2 · 3 · 4 · 5 · 6. That is where the number 13,983,816
• For the poker problem, the quotient is 52 · 51 · 50 · 49 · 48
divided by 1 · 2 · 3 · 4 · 5, which leads to 2,598,960 diﬀerent
poker hands. Note that since only four of these hands are
royal flushes, the probability of being dealt such a hand is
4/2,598,960 = 1/649,740, or about three royal flushes out of
every two million hands dealt.
• You can solve the handshake problem in your head: 14 · 13
divided by 1 · 2 is 91.
A Four-Mile-High Stack of Cards
The idea of randomly selecting the telephone number of an unknown
person as an aid in coming to grips with the tiny probability of win-
ning the lottery is not the only possible example. Here is another
We begin with the observation that a deck of cards placed on the
table is about an inch thick. It would take about 270,000 decks of
another way of picturing this small probability is oﬀered in Chapter 83.