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I. Pascal's Triangle

Pascal's triangle, of which the first eleven rows are shown below, gives the coefficients of binomial expansions (a + b)n.

The mth entry of the nth row of Pascal's triangle is given by

 (1)

for nonnegative n and m, and we have the Pascal formula

 (2)

The sum of the entries in the nth row is

II. Pascal's Simplices

Pascal's triangle is composed of binomial coefficients, each the sum of the two numbers above it to the left and right. Trinomial coefficients, the coefficients of the expansions (a + b + c)n, also form a geometric pattern. In this case the shape is a three-dimensional triangular pyramid, or tetrahedron. Each horizontal cross section ("row") of Pascal's tetrahedron is a triangular array of numbers, and the sum of three adjacent numbers in each row gives a number in the following row. (Try stacking up tennis balls or oranges to understand the geometry; each entry (or orange) rests on three others, as in the illustration of a tetrahedral packing to the right.) The first seven rows are shown below in cross section. Notice that each of the three faces of the tetrahedron is Pascal's triangle itself.

As the sum of the entries in the nth row of Pascal's triangle is 2n, the sum of the entries in the nth row of Pascal's tetrahedron is 3n, since this sum is simply

We can work out an explicit formula for the trinomial coefficients by treating a trinomial a + b + c as two nested binomials; first we expand it as the binomial a + (b + c) and then expand b + c inside:

 (3)

Thus the value of Pascal's tetrahedron located at (nm, mk, k) is the coefficient of anm bmk ck:

The trinomial coefficients satisfy the following additive relation:

As binomial coefficients form a triangle and trinomial coefficients form a tetrahedron, higher polynomial coefficients form higher dimensional objects. Quartic coefficients comprise a pentatope, the four-dimensional analogue of the tetrahedron. The three-dimensional cross sections of Pascal's pentatope are tetrahedrons, and the first five are given below in cross sections.

In general, we would like to understand the d-nomial, or multinomial, coefficients in the expansion of . They comprise the d-dimensional simplex of Pascal-generated numbers. The sum of all these coefficients, for given d and n, is dn. An explicit form can be found inductively. Let denote the coefficient of in the multinomial expansion of , where

Theorem 1  Multinomial coefficients have the explicit form

Proof. We have shown above that the statement holds for d = 3. Inductively, we assume it holds for some d ≥ 3 and show that it holds for d + 1. The coefficient of in

is (at m = nnd + 1 in the summation)

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We now change variables to obtain the additive relation for multinomial coefficients. Let m1 = n, and, for each 2 ≤ jd + 1, let mj = mj – 1nj – 1. In other words,

This enables us to write each multinomial coefficient as a product of binomial coefficients:

Let

Theorem 2  The d-nomial coefficients satisfy the additive relation

Proof. The additive relation can be written somewhat more succinctly by defining the function

We seek to prove that

 (4)

We prove the relation by induction. We have proved it above for d = 3. Assume that it holds for some d ≥ 3. Then we have

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From the expression of multinomial coefficients as a product of binomial coefficients, we notice, for example, that each linear row in the triangular cross sections of Pascal's tetrahedron is a row of Pascal's triangle, scaled by some integer. Additionally, each triangular row of the tetrahedral cross sections of Pascal's pentatope is a triangular cross section of Pascal's tetradedron, scaled by some integer. In general, each (d – 1)-simplicial row of the d-simplicial cross sections of Pascal's (d + 1)-simplex is a (d – 1)-simplicial cross section of Pascal's d-simplex, scaled by some integer.

References