# Fun Math Project #4 Pascal's Triangle

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Fun Math Project #4: Pascal’s Triangle
Pascal’s triangle is named after a seventeenth century French mathematician, Blaise Pascal (1623 – 1662),
independently discovered during the eleventh century by both the Persians and the Chinese. One of the earliest
displays of Pascal’s triangle dates back to 1261 to the work of Yang Hui. Yang Hui, a Chinese mathematician,
managed to construct the triangle as far as the 6th row. The Chinese call the numbers Yanghui’s Triangle because
Chu Shih-Chieh in 1303 gave credit to Yanghui, who worked about 40 years before him. There is evidence that
this number triangle was also familiar to the Arab astronomer, poet, and mathematician, Omar Khayyam, as early
as the eleventh century. This number triangle concept moved from China and Arabia to Europe, where many
mathematicians also contributed to the development of this triangle. In some Italian works, the array is called
Tartaglia’s Triangle, named for the Italian algebraist Tartaglia who published the numbers in 1556, over a
century before Pascal. In 1655, Blaise Pascal published a book where he developed many of the triangle’s
properties and applications.

The triangular arrangement of numbers shown below is known as Pascal’s triangle.
1
1       1
1     2       1
1        3       3      1
1      4      6       4      1
?       ?        ?       ?      ?      ?

Question 1: Find the six numbers designated by the question marks.

One of the uses of Pascal’s triangle is to obtain the coefficients of the terms involved in a binomial expansion:
1       Row 1: ( a + b) 0 = 1
1        1       Row 2: ( a + b)1 = 1a + 1b
1         2       1        Row 3: ( a + b) 2 = 1a 2 + 2ab + 1b 2
1        3        3        1       Row 4: ( a + b)3 = 1a 3 + 3a 2b + 3ab 2 + 1b 3
1        4         6       4        1       Row 5: ( a + b) 4 = 1a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + 1b 4
?        ?         ?       ?        ?       ?

Question 2: What patterns do you observe in Pascal’s triangle compared to patterns you see in the expansion of a
binomial?

Question 3: What is the expansion of ( a + b)5 involving Row 6?

Pascal’s triangle can be used in problem solving too. Consider the following problem:

Question 4: How many different downward paths are there from A to B in the grid below? The path must be
downward and travel on the lines.
A

B