Pascal's Pyramid

High Quality Content by WIKIPEDIA articles! 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.Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row 0, and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.

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High Quality Content by WIKIPEDIA articles! 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.Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row 0, and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row.