Pascals Triangle

Pascal’s triangle is a triangular array of binomial coefficients that arises in probability theory, combinatorics, and algebra, among other areas of mathematics. It is named after the French mathematician Blaise Pascal in much of the Western world, despite the fact that other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

The rows of Pascal’s triangle are traditionally enumerated from the top down, starting with row n=0 at the very beginning (the 0th row). Typically, the numbers in each row are staggered in relation to the numbers in the adjacent rows, with the first entry in each row starting with k=0 and continuing to the last entry in each row beginning with k=1. In order to construct the triangle, the following steps should be followed: 

There is a single non-zero entry 1 in row 0 (the topmost row) that is unique. For each subsequent row, each entry is constructed by multiplying the number above and to the left by the number above and to the right, with blank entries being treated as 0. Example: The first (or any other) row begins with the number 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row (or any other row).

Combinations

The calculation of combinations is the second application of Pascal’s triangle that can be useful in many situations. If you want to know how many different combinations of n things can be taken at a time (known as n choose k), the equation can be used to find the answer.

The formula for a cell of Pascal’s triangle, on the other hand, is the same. Rather than having to perform the calculation, one can simply look up the appropriate entry in the triangular database. For example, if we start with a row number of zero and then move down to the first entry in a row number of zero, the answer will be found at entry k in row n. Consider the following scenario: a basketball team has ten players and wants to know how many different ways there are to select eight of them. This is the answer: entry 8 in row 10, which is number 45; that is, 10 choose 8 equals number 45.

Relation to binomial distribution and convolutions:

Because it is multiplied by two and divided by 2n, the nth row of Pascal’s triangle is transformed into the binomial distribution in the symmetric case where p=1/2. According to the central limit theorem, as the number of observations increases, this distribution approaches the normal distribution. Stirling’s formula can also be used to demonstrate this by applying it to the factorials involved in the formula for combinations.

There are two ways in which this is related to the operation of discrete convolution. Because polynomial multiplication is an exact match for discrete convolution, repeatedly convolving the sequence {….,0,0,1,1,1…} with itself corresponds to taking powers of x+1 and, as a result, to generating the rows of the triangular pyramid. For example, To put it another way, repeatedly convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for an n-person sample consisting of n independent copies of that variable; this is the exact situation to which the central limit theorem applies, and thus results in the normal distribution being obtained in the limit.

Properties of Pascal’s triangle:

• Each number is made up of the sum of the two numbers immediately above it.
• The numbers on the outside are all one.
• The triangle has a symmetrical shape.
• The counting numbers are depicted on the first diagonal.
• Taking the sums of the columns, we get the powers of two.
• Each row contains the digits of the powers of 11 in descending order.

Logic of Pascal’s triangle:

Because it makes use of the concept of binomial coefficient, using a function is the most efficient method of printing Pascal’s triangle in C. However, the alternative source code provided below does not include any user-defined functions. Instead, it consists of a series of loops that print Pascal’s triangle in the standard format. 

The program’s source code is made up of six integer-type variables, which are denoted by the letters x, y, n, a, z, and s. A total of four variables have been defined: the variables x, y, and z are used to control the for() loop, the integer n stores the limit of Pascal’s triangle that has been entered by the user, and the variable s is used to print the space in the triangle.

Conclusion:

 It is a triangular arrangement of numbers used in algebra to determine the coefficients of any binomial expression, such as (x + y)n, and it is also known as Pascal’s triangle. Pascal’s triangle is a triangular array of binomial coefficients that arises in probability theory, combinatorics, and algebra, among other areas of mathematics. Pascal’s triangle also consists of various properties like, the triangle has a symmetrical shape. The counting numbers are depicted on the first diagonal. Taking the sums of the columns, we get the powers of two. Each row contains the digits of the powers of 11 in descending order.