1

11

121

1331

14641

15101051

1615201561

The triangle is named after French mathematician

B

LAISE

P

ASCAL

(1623–62), who used it extensively in

his development of

PROBABILITY

theory. However, Pas-

cal never claimed to have invented the triangle. It

appears, for example, in C

HU

S

HI

-

CHIEH

’s 1303 text

Siyuan Yujian (The precious mirror of the four ele-

ments), and tables of the first few rows of the triangle

appear in ancient Arab texts.

To motivate the construction of the triangle, one

can ask the following question:

If, in a rectangular grid, one is permitted to

move from cell to cell only by taking eastward

or southward steps, how many distinct routes

are there from the top left cell to any other

given cell?

For example, to move to the circled cell shown, one

could take three steps east followed by two steps south

(which we shall record as EEESS), or, alternatively, one

step south, two steps east, one step south, followed by

a final step east (SEESE). In fact, any arrangement of

three Es and two Ss corresponds to a possible path, and

one can check that there are 10 such combinations of

five letters in all. This shows that there are 10 routes to

the circled cell shown. It is convenient to assign the

number one to the top left cell. (Starting at the top left

cell there is only one course of action needed to reach

that cell, namely, no action.)

Pascal’s triangle 383

In 1642 Blaise Pascal invented and built the world’s first calculating machine. (Photo courtesy of the Science Museum, London/Topham-

HIP/The Image Works)