In the module, "Combinations and Their Sums," ibn Mun‘im’s pattern is also used to compress rather than to expand. A surprising range of problems can be simplified by this means.

For example, I ask my students to count the number of different possible sequences of wins and losses in the baseball World Series, assuming that the Detroit Tigers win the Series. Since the Series winner is the first to win 4 games, the length of the Series ranges from 4 to 7 games. In each case, the Tigers would need to win not only the final game played but any 3 of the preceding games, with the number of preceding games ranging from 3 to 6. Thus, it was easy to construct an exercise that leads students to see that the number of possibilities is:

\[\left(\begin{array}{c} 3\\3\end{array}\right)+\left(\begin{array}{c} 4\\3\end{array}\right)+\left(\begin{array}{c} 5\\3\end{array}\right)+\left(\begin{array}{c} 6\\3\end{array}\right).\]

The students are then asked to figure out how to use ibn Mun‘im’s pattern to compress this into a single coefficient, namely \(\left(\begin{array}{c} 7\\4\end{array}\right)=35\) sequences. This shortcut to the answer surprises many of them, in part because most Series do not last 7 games.

As another example of compression, I ask my students to count the total number of gifts exchanged in “The Twelve Days of Christmas” according to the traditional song:

On the 1st day of Christmas my true love gave to me: A Partridge in a Pear Tree.

On the 2nd day of Christmas my true love gave to me: Two Turtle Doves, and A Partridge in a Pear Tree.

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On the 12th day of Christmas my true love gave to me: Twelve Drummers Drumming, Eleven Pipers Piping, Ten Lords A-leaping, Nine Ladies Dancing, Eight Maids A-milking, Seven Swans A-swimming, Six Geese A-laying, Five Golden Rings, Four Calling Birds, Three French Hens, Two Turtle Doves, and A Partridge in a Pear Tree.

Here, the students must compress repeatedly and on two levels. First, they discover that on any given day, the number of gifts can be found by a combination involving a choice of two items. For instance, on Day 12 the gifts total:

\[12+11+\cdots+1= \left(\begin{array}{c} 12\\1\end{array}\right)+\left(\begin{array}{c} 11\\1\end{array}\right)+\cdots +\left(\begin{array}{c} 1\\1\end{array}\right)=\left(\begin{array}{c} 13\\2\end{array}\right).\]

Based on this, the students are able to conclude that over the 12 days, the grand total is:

\[\left(\begin{array}{c} 2\\2\end{array}\right)+\left(\begin{array}{c} 3\\2\end{array}\right)+\cdots +\left(\begin{array}{c} 13\\2\end{array}\right)=\left(\begin{array}{c} 14\\3\end{array}\right)=364\,\,{\rm gifts,}\]

with the final calculation based on the standard factorial formula. Thus, instead of needing to calculate many different numbers and then their total, the students are able to count the gifts by a single calculation involving factorials. That the work can be compressed into a single binomial coefficient is very surprising, especially since the numbers 3 and 14 bear no obvious relation to the original statement of the problem. Such examples illustrate in a dramatic way the utility of ibn Mun‘im’s insight.

The students go on to compare ibn Mun‘im’s arithmetical triangle with later ones from Chu Shi-Chieh (China, 1303) and Blaise Pascal (France, 1665) (on both, see Edwards 1987). Interestingly, for Pascal and others in Europe investigating combinatorics, the major stimulus was to answer questions arising from games of dice. By contrast, in the medieval Muslim world such games were rarely played because Islam disapproves of wagering and other kinds of financial speculation. Instead, combinatorial research among Muslims was driven by their interest in Arabic, which they considered sacred because it is the language of the *Qur’?n.* Their goal was to count the number of ways that Arabic letters and sounds can be combined; this was, in fact, the larger project for which ibn Mun‘im developed his triangle. In medieval India, combinatorics came to be of special interest to mathematicians of the Jaina religion, who studied combinations of senses, of philosophical categories, etc. (Joseph 2000: 253-255; Katz 1996: 99-101; Katz 2009: 250-252). All of these applications provide wonderful raw material for student activities. Comparing such examples (the fifth and final point of strategy mentioned above) allows us to impart to our students the lesson that mathematics takes diverse forms in different parts of the world because it is shaped by culture.

Download the module that includes the exercises described on this page, Combinations and Their Sums (Elementary Statistics, Finite Mathematics), and the follow-up module, Binomial Coefficients and Subsets (Finite Mathematics).