Pascal Triangle

In our daily life we use widely the mathematics branch called combinatorial analysis. This one studies so-called finite sets. The set consisting of n elements is called n-element one. However we can chose k elements from n-element set. Each k-element part of the n-element set is called combination from given n elements by k. One of the problems of combinatorial analysis is to find a number of combinations of n elements by k. Usually this number is marked as .

Let us calculate Let us begin from . But what means the 0-element set? It means that the set has not elements. This set is called an "empty" set. It is clear that there exists only one combination of n elements by 0, that is = 1.

Let us consider a set consisting of 3 elements: a pencil, a pen and a lasting. Let us calculate for this case. It is clear that = 1.

Let us calculate . It is clear there exist only3 1-element parts for this case, that is = 3.

For the case k = 2 also there exist only 3 2-element parts, that is = 3.

At least for the case k = 3 there exists only 1 3-element part, that is = 1.

But how match is a number of all possible parts of n-element set. For our example we have:

In combinatorial analysis it was proved the following general result:

(1)

In mathematics the numbers are called binomial coefficients. From secondary school we know so-called binomial formula, which has the following form:

(2)

where are called the binomial coefficients. Also the formula (2) is called Newton's formula.

In fact, Newton used this formula in his mathematical investigations. But historically this name is not correct because formula (2) was well known by the Arabian mathematicians long before Newton. The famous French mathematician Pascal suggested a very simple way of the binomial coefficients calculation by means of their disposition in the form of certain array called the arithmetical square or Pascal triangle.

Let us consider the so-called the right Pascal triangle, which presents by itself the following array of numbers:

The rows of the Pascal triangle are numbered from the top down to the lower row. The binomial coefficients

are called the zero row. The n-row begins with the binomial coefficient = 1 (n =0, 1, 2, 3, ... ).

The columns are numbered from left to right; the leftward extreme column consisting of the only member ( = 1) is called the zero column. The n-th column involves the binomial coefficients

where

Note that the binomial coefficient is on intersection of n-th column and k-th row. The binomial coefficient are connected with the following correlation:

(3)

Let us consider the Pascal triangle presented in a number form.

0-Pascal Triangle

1111111111
       123456789
       1361015212836
       141020355684
       15153570126
       162156126
       172884
       1836
       19
       1
1248163264128256512

We can see that the top row (the 0-row) of Pascal triangle consists of 1's and all diagonal binomial coefficients are equal to 1. Each binomial coefficient inside Pascal triangle are calculated according to (3). If we sum all binomial coefficients of n-th column we get the binary number 2n that corresponds to formula (2) (see bottom row of Pascal triangle). Thus, if we move along the bottom row starting since 0-th column we will get the binary series: 1, 2, 4, 8, 16, ... .

We will name the classical Pascal Triangle as the 0-Pascal Triangle. The meaning of a such definition will be revealed in the next page of our Museum.

The binomial coefficients and the Pascal triangle are widely practiced in different branches of mathematics. "This array has a set of wonderful properties, - wrote the famous mathematician J. Bernully, - just now we have shown that it conceals the essence of the connection theory, but those who are closer to geometry know that it hides a lot of fundamental secrets of other branches of mathematics".

But there arises a question: of what relation has Pascal Triangle to the subject of our Museum, that is, to Fibonacci numbers and golden section? We will show at the next pages of our Museum that there exists a deep connection between Pascal triangle and Fibonacci numbers. Moreover, Pascal Triangle plays a very important role for creation of special mathematics, the Mathematics of Harmony, which is intended for simulation of harmonies processes in nature.