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

comes from.

• 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

one.1

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

1Yet

another way of picturing this small probability is oﬀered in Chapter 83.