Section 4.3 The Arithmetic Triangle (Pascal's Triangle)
You may well have been introduced to the following triangle (of which we show only the first six rows here):
In the \(n\)th row of the triangle (starting with row \(0\)), the binomial coefficients \(\binom{n}{k}\) appear, with \(k\) going from \(0\) to \(n\) as we proceed across the row.
The combinatorial identities shown in Example 4.2.7 and Exercise 4.2.10.1 provide techniques for calculating the actual values of the entries in a particular row, from the entries in the rows above it (we also note that the first and last entries in any row are \(1\)s). The second of these identities provides the simplest calculation: each entry is the sum of the entry above and to the left, with the entry above and to the right.
Below we show the calculated values for the part of the triangle listed above.
This triangle has been used in various forms for many centuries. The Moroccan mathematician Ahmad ibn Mun'im al-'Abdarī (11??—1228) drew the first \(10\) rows in his calculations of the number of tassels that could be produced using combinations of \(10\) colours of silk thread (see Example 4.2.7), in the \(1200\)s. The Indian poet Halayudha (10th century CE) in the \(900\)s drew it as a tool in working out the numbers of various kinds of poetic metres that he was interested in (with a specified number of syllables per line, a particular number of which had to be long or short). In the book Siyuan Yuchian (“Jade Mirror of the Four Unknowns”), produced in \(1303\text{,}\) the Chinese mathematician Zhu Shijie (1249—1314) drew this triangle (and attributed it to much older sources). Although the first documented use by Blaise Pascal (1623—1662) of the triangle was in \(1654\text{,}\) it is still usually (Eurocentrically) generally referred to as “Pascal's triangle”. The “Arithmetic triangle” is a more neutral term that is still used by some sources, and was in fact the term used by Pascal in his 1654 work Treatise on the Arithmetic Triangle, but this term is less likely to be recognised.