Pascal's Pyramid - Relationship With Pascal's Triangle

Relationship With Pascal's Triangle

It is well known that the numbers along the three outside edges of the nth Layer of the tetrahedron are the same numbers as the nth Line of Pascal's triangle. However, the connection is actually much more extensive than just one row of numbers. This relationship is best illustrated by comparing Pascal's triangle down to Line 4 with Layer 4 of the tetrahedron.

Pascal's triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

Tetrahedron Layer 4
1 4 6 4 1
4 12 12 4
6 12 6
4 4
1

Multiplying the numbers of each line of Pascal's triangle down to the nth Line by the numbers of the nth Line generates the nth Layer of the Tetrahedron. In the following example, the lines of Pascal's triangle are in italic font and the rows of the tetrahedron are in bold font.

1
× 1 =
1

1 1
× 4 =
4 4

1 2 1
× 6 =
6 12 6

1 3 3 1
× 4 =
4 12 12 4

1 4 6 4 1
× 1 =
1 4 6 4 1

The multipliers (1 4 6 4 1) compose Line 4 of Pascal's triangle.

This relationship demonstrates the fastest and easiest way to compute the numbers for any layer of the Tetrahedron without computing factorials, which quickly become huge numbers. (Extended precision calculators become very slow beyond Tetrahedron Layer 200.)

If the coefficients of Pascal's triangle are labeled C(i,j) and the coefficients of the Tetrahedron are labeled C(n,i,j), where n is the layer of the Tetrahedron, i is the row, and j is the column, then the relation can be expressed symbolically as:

C(i,j) × C(n,j) = C(n,i,j) i = 0 to n, j = 0 to i

Read more about this topic:  Pascal's Pyramid

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