Pascal's Pyramid

Precision

Any of you come across the multinomial theorem?
Is it ever used in solving statistical problems?

Pascal's Tetrahedron (Pascal's Pyramid)...

1

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1
1 1

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1
2 2
1 2 1

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1
3 3
3 6 3
1 3 3 1

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1
4 4
6 12 6
4 12 12 4
1 4 6 4 1

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1
5 5
10 20 10
10 30 30 10
5 20 30 20 5
1 5 10 10 5 1

Kevster

I have been using this frequently for the past two years. It is used for statistics and probability. For example, the Binomial Probability formula.

Ah! - Excellent! I found an excellent link that shows how it's used:

http://www.krysstal.com/binomial.html


...I couldn't be arsed explaining myself :rolleyes:

Mellor

The one I was familar with was the same as all the bottom lines of your triangles,

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1

What are the extra numbers in yours.
We used to in probability and in alegbra, (x+y)^n is the nth line on the triangle. with x^n reducing, y^0 increasing

Precision

The Binomial Probability is used for calculating the probability when throwing a 2 sided coin a number of times.

Trinomial probability would be used for calculating the probability when throwing an object that could land on one of 3 possible sides.

Tetranomial probability would be used for calculating the probability when throwing a tetrahedron dice (4 sided dice, d4).
Hexanomial probability would be used for calculating the probability when throwing a cubic dice (6 sided dice, d6).
Octanomial probability would be used for calculating the probability when throwing a octahedron dice (8 sided dice, d8).

Likewise, a d12 is a dodecahedron dice (12 sided dice) and a d20 is a icosahedron dice (20 sided dice).

Say if a d20 dice was thrown a large number of times, although any arrangement of numbers is possible, some overall combinations of numbers are far more likely to occur than other combinations.

I am not sure though whether the mathematics would get to difficult. I think there would be a lot of recursion in the maths.

Precision

Pascal's Triangle is a two dimenional layout of the coefficients that are used in the binomial theorem.

The Binomial Theorem
Here there are two elements, a and b.
The expansion of (a + b)^n for n = 0, 1,…, 5:

(a + b)^0 = 1
(a + b)^1 = 1(a) + 1(b)
(a + b)^2 = 1(a.a) +2(a.b) + 1(b.b)
(a + b)^3 = 1(a.a.a) +3(a.a.b) + 3(a.b.b) + 1(b.b.b)
(a + b)^4 = 1(a.a.a.a) +4(a.a.a.b) + 6(a.a.b.b) +4(a.b.b.b) + 1(b.b.b.b)
(a + b)^5 = 1(a.a.a.a.a) +5(a.a.a.a.b) +10(a.a.a.b.b) +10(a.a.b.b.b) +5(a.b.b.b.b) +1(b.b.b.b.b)

Pascal's Tetrahedron is a three dimenional layout of the coefficients that are used in the trinomial theorem.

The Trinomial Theorem
Here there are three elements, a, b and c.
The expansion of (a + b + c)^n for n = 0, 1,…, 3:

(a + b + c)^0 = 1
(a + b + c)^1 = 1(a) + 1(b) + 1(c)
(a + b + c)^2 = 1(a.a) +2(a.b) + 2(a.c) + 2(b.c) + 1(b.b) + 1(c.c)
(a + b + c)^3 = 1(a.a.a) +3(a.a.b) +3(a.b.b) +1(b.b.b) +6(a.b.c) +3(a.a.c)+3(a.c.c)+3(b.b.c)+3(b.c.c)+1(c.c.c)

Multinomial theorems, with a higher number of elements, would have a higher dimensional layout of coefficients.

The tetranomial theorem, is used when there are four elements, and its layout of coefficients is four dimensional.