Pascal's Triangle

Pascal’s Triangle is a famous triangular array of numbers with several interesting properties.  Each line of the triangle starts and ends with the number 1, and the numbers in between can be obtained by finding the sum of the two numbers above it.  The triangle can then be continued indefinitely.

                            

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

Pascal’s Triangle

Unfortunately, Pascal’s triangle is rarely mentioned in high school math curriculum, except as a footnote for expanding binomials, because the coefficients for the expanded form of (x + y)ⁿ match up with the rows of Pascal’s Triangle.  For example, (x + y)² = x² + 2xy + y² (1, 2, 1), (x + y)³ = x³ + 3x²y + 3xy² + y³ (1, 3, 3, 1), and (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ (1, 4, 6, 4, 1).  But there are many other unique properties to the triangle.

First of all, the rows of Pascal’s triangles add up to a power of 2.  For example, the numbers in the third row (1, 2, 1) add up to 4 which is 2², the numbers in the fourth row (1, 3, 3, 1) add up to 8 which is 2³, and the numbers in the fifth row (1, 4, 6, 4, 1) add up to 16 which is 2⁴.

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

The rows of Pascal’s Triangle add up to powers of 2.

Second, the diagonals of Pascal’s Triangle represent totals for different simplexes (lines, triangles, tetrahedrons, etc.).  For example, the third diagonal row (1, 3, 6, 10, 15, etc.) are triangular numbers, because objects in these amounts can form equilateral triangles, and the fourth diagonal row (1, 4, 10, 20, etc.) are tetrahedral numbers, because of objects in these amounts can form tetrahedrons (triangular pyramids).

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

The diagonals of Pascal’s Triangle represent totals for different simplexes.

Triangular Numbers: 1, 3, 6, 10, etc.

Tetrahedral Numbers: 1, 4, 10, 20, etc.

Third, another diagonal of a different slope in the numbers of Pascal’s triangle add up to different numbers in the Fibonacci sequence.  As you might recall, the numbers in the Fibonacci sequence can be obtained by adding the two numbers before it, so starting with two ones, it is 1, 1, 2, 3, 5, 8, 13, etc.  Numbers in the pictured diagonals (and parallel diagonals) of Pascal’s triangle add up to the numbers in the Fibonacci sequence.  For example, 1 = 1, 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 1 + 3 + 1 = 5, 1 + 4 + 3 = 8, 1 + 5 + 6 + 1 = 13, etc.

						1
					1		1
				1		2		1
			1		3		3		1
		1		4		6		4		1
	1		5		10		10		5		1
1		6		15		20		15		6		1

These Pascal Triangle diagonals add up to different numbers in the Fibonacci sequence.

Fourth, the numbers in the Pascal’s triangle represent the number of combinations that are possible in a set, which is useful in statistics.  For example, if you wanted to know how many different possible two-card hands you can be dealt from a deck of five cards, it would be ₅C₂ = 10, which is 5 + 1 = 6^th row, and 2 + 1 = 3^rd number in Pascal’s Triangle.  (If the five cards are A, B, C, D, and E, then all the possible two-card hands are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE, which makes ten total possibilities.)  The number of different possible four-card hands you can be dealt from a deck of six cards would be ₆C₄ = 15, which is the 6 + 1 = 7^th row, and 4 + 1 = 5^th number in Pascal’s Triangle.

0C0

1C0

1C1

2C0

2C1

2C2

3C0

3C1

3C2

3C3

4C0

4C1

4C2

4C3

4C4

5C0

5C1

5C2

5C3

5C4

5C5

6C0

6C1

6C2

6C3

6C4

6C5

6C6

Combination Sets in Pascal’s Triangle

Finally, as mentioned in an earlier blog post, the number of endpoints, line segments, faces, and solids of each multi-dimensional simplex corresponds to a row in Pascal’s Triangle.  For example, the 1-D simplex (line segment) is comprised of 2 endpoints and 1 line segment (2, 1); the 2-D simplex (triangle) is comprised of 3 endpoints, 3 line segments, and 1 face (3, 3, 1); the 3-D simplex (tetrahedron) is comprised of 4 endpoints, 6 line segments, 4 faces, and 1 solid (4, 6, 4, 1), etc.

3D Simplex: Tetrahedron

Comprised of 4 endpoints, 6 line segments, 4 faces, and 1 solid

Pascal’s Triangle has some amazing unique properties.  It can be applied to the expansion of binomials, powers of two, sequences of numbers, statistics, and properties of simplexes.  It truly is a wonder in mathematics!