Pascal's Triangle

By de Mestre, Neville | Australian Mathematics Teacher, Fall 2010 | Go to article overview

Pascal's Triangle


de Mestre, Neville, Australian Mathematics Teacher


Consider the pattern of numbers shown in Figure 1.

[FIGURE 1 OMITTED]

The first row has the element 1, while the second row has two ones symmetrically staggered either side of the number in the first row so that an equilateral triangular pattern is being formed. The third row again has ones in the outside positions but the central number is the sum of the two staggered numbers just above it. The fourth row continues this process and is therefore

1    1 + 2 1 + 2    1

This triangular pattern is known as Pascal's triangle, named after the French mathematician Blaise Pascal (1623-1662). Although Pascal's name is associated with it today, it was known many years beforehand. The Chinese mathematician Chu Shi-kie published it in China in 1303 and he claimed that even then it was an ancient fact.

See if your students can discover the next three rows for themselves. Then ask them to find the sum of the numbers in each row. Can they deduce the sum for the tenth row?

When a coin is tossed it will come down showing a head (H) or a tail (T) on landing. When two coins are tossed at the same time, the result is either HH, HT, TH, or TT. In tabular form we could write this as:

No tail      One tail     Two tails

HH           HT, TH       TT
1            2            1

This is the third row of Pascal's triangle. Now ask your class to show the tabular results for tossing three coins at the same time, then four coins and perhaps five.

In an earlier Discovery article (de Mestre, 2008), I investigated patterns with ones. Consider

[11.sup.0] = 1

[11.sup.1] = 11

[11.sup.2] = 121

[11.sup.3] = 1331

[11.sup.4] = 14641

Note that the triangle has been generated again. However it only happens for a limited number of rows. See if your students can find out when it stops.

Pascal's triangle can also be seen in a square grid such as the 5 x 5 array shown in Figure 2 , or one even longer.

[FIGURE 2 OMITTED]

Consider the cells as stepping paths. If a person starts at S there is only one shortest path they can take to reach each of the cells in the first row. But to reach a cell in the second row from S, the number of equal shortest paths are the counting numbers as shown. To reach the third row from S your students should try to discover the numbers shown in Figure 2. These are the triangular numbers, and for the fourth row the tetrahedral numbers. Pascal's triangle can now be seen along the diagonal cells from left to right.

The above construction can be extended very easily by your students using a spreadsheet application, such as Excel.

Write 1 in cell A1 and fill across row 1 for as far as you like.

Next fill down column A, again obtaining all ones.

Write "=A2+B1" in cell B2 and fill down the rest of column B or to where you stopped filling down. …

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