In mathematics, Pascal's pyramid is a three-dimensional arrangement of the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial coefficients that appear in the binomial expansion and the binomial distribution. The binomial and trinomial coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.
Structure of the tetrahedron Because the tetrahedron is a three-dimensional object, displaying it on a piece of paper, a computer screen, or other two-dimensional medium is difficult. Assume the tetrahedron is divided into a number of levels, floors, slices, or layers. The top layer (the apex) is labeled "Layer 0". Other layers can be thought of as overhead views of the tetrahedron with the previous layers removed. The first six layers are as follows:
The layers of the tetrahedron have been deliberately displayed with the point down so that they are not individually confused with Pascal's triangle.
Overview of the tetrahedron There is three-way symmetry of the numbers in each layer. The number of terms in the nth layer is the (n + 1)th triangular number: ( n + 1 ) ( n + 2 ) 2 {\displaystyle {\frac {(n+1)(n+2)}{2}}} . The sum of the values of the numbers in the nth layer is 3n. Each number in any layer is the sum of the three adjacent numbers in the layer above. Each number in any layer is a simple whole number ratio of the adjacent numbers in the same layer. Each number in any layer is a coefficient of the trinomial distribution and the trinomial expansion. This non-linear arrangement makes it easier to: display the trinomial expansion in a coherent way; compute the coefficients of the trinomial distribution; calculate the numbers of any tetrahedron layer. The numbers along the three edges of the nth layer are the numbers of the nth line of Pascal's triangle. And almost all the properties listed above have parallels with Pascal's triangle and multinomial coefficients.
Trinomial expansion connection The numbers of the tetrahedron are derived from the trinomial expansion. The nth layer consists of all the coefficients when the trinomial A + B + C {\displaystyle A+B+C} is raised to the nth power. The nth power of the trinomial is expanded by repeatedly multiplying the trinomial by itself:
Each term in the first expression is multiplied by each term in the second expression; and then the coefficients of like terms (same variables and exponents) are added together. Here is the expansion of (A + B + C)4:
Writing the expansion in this non-linear way shows the expansion in a more understandable way. It also makes the connection with the tetrahedron obvious−the coefficients here match those of layer 4. All the implicit coefficients, variables, and exponents, which are normally not written, are also shown to illustrate another relationship with the tetrahedron. (Usually, "1A" is "A"; "B1" is "B"; and "C0" is "1"; etc.) The exponents of each term sum to the layer number (n), or 4, in this case. More significantly, the value of the coefficients of each term can be computed directly from the exponents. The formula is (x+y+z)!/x!y!z! , where x, y, z are the exponents of A, B, C, respectively, and "!" is the factorial, i. e.: n ! = 1 ⋅ 2 ⋅ 3 ⋯ n {\displaystyle n!=1\cdot 2\cdot 3\cdots n} . The exponent formulas for the 4th layer are:
The exponents of each expansion term can be clearly seen and these formulae simplify to the expansion coefficients and the tetrahedron coefficients of layer 4.
Trinomial distribution connection The numbers of the tetrahedron can also be found in the trinomial distribution. This is a discrete probability distribution used to determine the chance some combination of events occurs given three possible outcomes−the number of ways the events could occur is multiplied by the probabilities that they would occur. The formula for the trinomial distribution is:
where x, y, z are the number of times each of the three outcomes does occur; n is the number of trials and equals the sum of x+y+z; and PA, PB, PC are the probabilities that each of the three events could occur. For example, in a three-way election, the candidates got these votes: A, 16 %; B, 30 %; C, 54 %. What is the chance that a randomly selected four-person focus group would contain the following voters: 1 for A, 1 for B, 2 for C? The answer is:
The number 12 is the coefficient of this probability and it is number of combinations that can fill this "112" focus group. There are 15 different arrangements of four-person focus groups that can be selected. Expressions for all 15 of these coefficients are:
The numerator of these fractions (above the line) is the same for all expressions. It is the sample size−a four-person group−and indicates that the coefficients of these arrangements can be found on layer 4 of the tetrahedron. The three numbers of the denominator (below the line) are the number of the focus group members that voted for A, B, C, respectively. Shorthand is normally used to express combinatorial functions in the following "choose" format (which is read as "4 choose 4, 0, 0", etc.).
But the value of these expression is still equal to the coefficients of the 4th layer of the tetrahedron. And they can be generalized to any layer by changing the sample size (n). This notation makes an easy way to express the sum of all the coefficients of layer n:
Addition of coefficients between layers The numbers on every layer (n) of the tetrahedron are the sum of the three adjacent numbers in the layer (n−1) "above" it. This relationship is rather difficult to see without intermingling the layers. Below are italic layer 3 numbers interleaved among bold layer 4 numbers:
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