Pascal's pyramid
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In mathematics, Pascal's pyramid is a three dimensional generalization of Pascal's triangle. Just as Pascal's triangle gives coefficients for the terms of a binomial expansion, so Pascal's pyramid gives coefficients for a trinomial expansion.
Pascal's pyramid is also called Pascal's tetrahedron. A generalization to arbitrary dimension is sometimes called a Pascal's simplex.
The pyramid is constructed layer by layer, starting with the apex and working downward. The first few layers are as follows.
1 1 1 1 1
1 1 2 2 3 3 4 4
1 2 1 3 6 3 6 12 6
1 3 3 1 4 12 12 4
1 4 6 4 1
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Properties of Pascal's pyramid
Summing the numbers in each column of a layer of Pascal's pyramid gives the nth power of 111 in base infinity (i.e. without carrying over during multiplication), where n is the layer - 1.
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1 |1| | |1| | | | |1| | | | | | | 1| | | |
--- 1| |1 |2| |2| | |3| |3| | | | | 4| | 4| | |
1 ----- 1| |2| |1 |3| |6| |3| | | 6| |12| | 6| |
1 1 1 --------- 1| |3| |3| |1 |4| |12| |12| |4|
1 2 3 2 1 ------------- 1| | 4| | 6| | 4| |1
1 3 6 7 6 3 1 ----------------------
1 4 10 16 19 16 10 4 1
1110 1111 1112 1113 1114 = 151807041
1*108+4*107+10*106+16*105+19*104+16*103+10*102+4*101+1*100
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External links
- Pascal's Simplices (discusses Pascal's triangle, Pascal's pyramid, and more)pdc:Pascal's Pyramid