PASCAL’S TRIANGLE

Pascal's Triangle or Arithmetic Triangle, a pattern of numbers obtained by expanding successive powers of x + y, namely (x + y)1, (x + y)2, and so on, giving the array of coefficients shown in the figure.
Having infinitely many rows, and only two sides, this is not really a triangle. The rows are numbered n = 1, 2, ... from the top downwards; the entries in row n are the coefficients of the terms in the expansion of (x + y) n. These are called the binomial coefficients, (pronounced “n choose k”). Here n! (“n factorial”) means n × (n -1) × (n -2) × ... × 2 × 1 for n≥ 1. The expression gives the number of ways of choosing k objects from a set of n objects: hence the name. For example, the coefficient of x 2y2 in (x + y)4 is
Each entry in Pascal's triangle (apart from the 1s along the sides) is the sum of the two entries to its left and right in the preceding row. Using this fact, one can construct further rows of the triangle.
Pascal's triangle displays many other interesting numerical relationships and patterns. For instance, the sum of the entries in row n is 2n. Thus the sum of the entries in row 4 is 24 = 16. Furthermore, replacing even and odd terms in Pascal's triangle with 0 and 1 respectively, we get the following self-replicating pattern:
The next eight rows consist of two adjacent copies of this triangle, with an inverted triangle of 0s between them, and so on.

History

The 17th-century French mathematician and theologian Blaise Pascal studied the binomial coefficients in connection with probability theory and games of chance; for instance, if an unbiased coin is tossed n times, the number of ways of obtaining k heads is
so the probability of this event is dependent on this number. However, Pascal was not the first to discover this array. It appeared in 1527 on the front page of the Rechnung, an arithmetic book by the German mathematician and astronomer Peter Apian (1495-1552), and the Chinese mathematician Chu Shih-Chieh referred to it in 1303, in his book The Precious Mirror, as “the old method”. The triangle probably dates from about 1100, when the Persian poet and mathematician Omar Khayyam appears to refer to it in his Algebra.











Pascal's Triangle
Pascal’s triangle is a pattern of numbers that is significant in mathematics. It is obtained very simply. The first row contains 1 twice. Each row after that is obtained by placing the sum of adjacent numbers below and between those numbers, with 1 beginning and ending each row. The resulting pattern of numbers mirrors the numbers obtained when (x + y) is multiplied by itself repeatedly. The fully written-out form of (x + y) n is called its expansion. As the comparison above shows, the numbers making up Pascal’s triangle are those appearing in the terms making up the successive expansions of (x + y) n as n takes on successive values. Pascal’s triangle has applications in various areas of mathematics, including probability theory.