Pascal Triangle In our daily life we use widely the mathematics branch called 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 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 At least for the case But how match is a number of all possible parts of In combinatorial analysis it was proved the following general result:
In mathematics the numbers are called
where are called the 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 Let us consider the so-called the 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 The columns are numbered from left to right; the leftward extreme column consisting of the only member ( = 1) is called the zero column. The where Note that the binomial coefficient is on intersection of
Let us consider the Pascal triangle presented in a number form.
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 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 |