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number of distinct seatings of 10 you can form from a group of 100 guests, you would start by looking down the numbers to the left of the triangle until you found the row labeled 100. The triangle I supplied does not go down that far, but for now let’s pretend it does. The first number in row 100 tells you the number of ways you can choose 0 guests from a group of 100. There is just 1 way, of course: you simply don’t choose anyone. That is true no matter how many total guests you are choosing from, which is why the first number in every row is a 1. The second number in row 100 tells you the number of ways you can choose 1 guest from the group of 100. There are 100 ways to do that: you can choose just guest number 1, or just guest number 2, and so on. That reasoning applies to every row, and so the second number in each row is simply the number of that row. The third number in each row represents the number of distinct groups of 2 you can form, and so on. The number we seek—the number of distinct arrangements of 10 you can form—is therefore the eleventh number in the row. Even if I had extended the triangle to include 100 rows, that number would be far too large to put on the page. In fact, when some wedding guest inevitably complains about the seating arrangements, you might point out how long it would have taken you to consider every possibility: assuming you spent one second considering each one, it would come to roughly 10,000 billion years. The unhappy guest will assume, of course, that you are being histrionic.

In order for us to use Pascal’s triangle, let’s say for now that your guest list consists of just 10 guests. Then the relevant row is the one at the bottom of the triangle I provided, labeled 10. The numbers in that row represent the number of distinct tables of 0, 1, 2, and so on, that can be formed from a collection of 10 people. You may recognize these numbers from the sixth-grade quiz example—the number of ways in which a student can get a given number of problems wrong on a 10-question true-or-false test is the same as the number of ways in which you can choose guests from a group of 10. That is one of the reasons for the power of Pascal’s triangle: the same mathematics can be applied to many different situations. For the Yankees-Braves World Series example, in which we tediously counted all the possibilities for the remaining 5 games, we can now read the number of ways in which the Yankees can win 0, 1, 2, 3, 4, or 5 games directly from row 5 of the triangle:

1 5 10 10 5 1

We can see at a glance that the Yankees’ chance of winning 2 games (10 ways) was twice as high as their chance of winning 1 game (5 ways).

Once you learn the method, applications of Pascal’s triangle crop up everywhere. A friend of mine once worked for a start-up computer-games company. She would often relate how, although the marketing director conceded that small focus groups were suited for “qualitative conclusions only,” she nevertheless sometimes reported an “overwhelming” 4-to-2 or 5-to-1 agreement among the members of the group as if it were meaningful. But suppose you hold a focus group in which 6 people will examine and comment on a new product you are developing. Suppose that in actuality the product appeals to half the population. How accurately will this preference be reflected in your focus group? Now the relevant line of the triangle is the one labeled 6, representing the number of possible subgroups of 0, 1, 2, 3, 4, 5, or 6 whose members might like (or dislike) your product:

1 6 15 20 15 6 1

From these numbers we see that there are 20 ways in which the group members could split 50/50, accurately reflecting the views of the populace at large. But there are also 1 + 6 + 15 + 15 + 6 + 1 = 44 ways in which you might find an unrepresentative consensus, either for or against. So if you are not careful, the chances of being misled are 44 out of 64, or about two-thirds. This example does not prove that if agreement is achieved, it is random. But neither should you assume that it is significant.

Pascal and Fermat