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PASCAL’S TRIANGLE AND ITS REAL LIFE APPICATION POOJA K S THE AVILA COLLEGE OF EDUCATION ABSTACTThe Pascal’s triangle is a triangular array of the binomial coefficients. In the western world, it is named after French mathematician Blaise Pascal. The purpose of the study is to make the students become aware of the topic and help the students to produce the first eleven rows of Pascal’s triangle. INTRODUCTIONThe Pascal’s triangle is a triangular array of the binomial coefficients. In the western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany and Italy.The rows of Pascal’s triangle are conventionally enumerated starting with row n=0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k=0 and are usually staggered relative to the numbers in the adjacent rows. Having the indices of both rows and columns start at zero makes it possible to state that the binomial coefficient nk appears in the nth row and kth columns of Pascal’s triangle. A simple construction of the triangle proceeds in the following manner. In row 0, the topmost row, the entry is 0 0 =1 the entry is in the 0th row and 0th column. Then to construct the elements of the following rows, add the number above and to the left with the number above and to the right of a given position to find the new value to place in that position. If either the number to the right or left is not present, substitute a zero in its place.The pattern of numbers that forms Pascal’s triangle was known well before Pascal’s time. Pascal’s innovated many previously unattested uses of the triangle’s numbers, uses he described comprehensively in what is perhaps the earliest known mathematical treatise to be specially devoted to the triangle, his trait du triangle arithmetique (1653). Centuries before, discussion of the numbers had arisen in the context of Indian studies of combinatorics and of binomial numbers and Greek’s study of figurate numbers. OBJECTIVES The main objective of the project “PASCAL’S TRIANGLE AND ITS REAL LIFE APPICATION” is to make students to recognize the integers, row and columns that comprise Pascal’s triangle The Pascal’s triangles enable to reproduce the first eleven rows of Pascal’s triangle by recalling number patterns given in the portion without having to look again at the original triangle. The real life application listed out in this project help the pupil to realize the importance of Pascal’s Triangle.HYPOTHESIS It assumes that the students will be able to understand the basic concept as how Pascal’s triangle is constructed. Also they may be able to acquire the Pascal’s triangle and they may develop the skill of representing the given situation I our daily life. SCOPE OF THE STUDY The Fibonacci SequenceFibonacci Sequence of numbers starts with 1, 1, and then the next number is added to the previous number to become the next number in the sequence and so on. This creates: 1,1,2,3,5,8,13,21,34, 55,89,144,233, etc . . . . This sequence can also be located in Pascal's Triangle Binomial ExpansionsThe expanded power of (a+b)n where a+b is any binomial and n is a whole number. The pattern is the coefficient of the expanded values follows the Pascal’s triangle according to the power. In this case the coefficient of the expanded follows that of 112 (121). Square Rings Place a token on each of six numbers surrounding a number on Pascal’s Triangle. What do you notice? The product of the surrounding number is a perfect square! Also the products of alternate triples are equal!For example: 1 x 5 x 6 = 1 x 10 x 3 = 30 and 3 x 10 x 4 = 4 x 10 x 3 = 120. Prime Number If the 1st element in a row is a prime number remembering the 0th element of every row is 1all the numbers in that row (excluding the 1’s) are divisible by it.For example: In row 7 1 7 21 35 21 7 1 7, 21 and 35 are all divisible by 7. The Probability of Heads or Tails Pascal originally discovered the properties of this triangle by thinking of problems posed by gamblers. Investigate what happens when a coin is flipped. If you flip a coin, it will come up either heads or tails.An equivalent could be said for flipping tails. Cover the first 1 at the top of Pascal’s Triangle (row 0) with a token to represent the flip of zero coins. Next, cover the numbers on the 1st row to represent the probability of getting a head on the toss of 1 coin. There are only 2 possibilities when tossing one coin and the probability of the coin coming up heads is 1 chance out of 2, or ?. The probability of any event is expressed as a fraction between 0 and 1 inclusive. Compare the number equivalent outcomes to the number in the 1st row of the triangle.One coin tossed Number of heads Number of equivalent outcomes 1st possible outcome: H 1 1 2nd possible outcome: T 0 1Suppose you toss 2 coins. Now cover the numbers in the 2nd row to represent the number equivalent outcomes when tossing 2 coins. The following are the possible outcomes Number of heads Number of equivalent outcome 1 1 2 0 1The probability of getting 2 heads from tossing 2 coins is: 1 chance out of 4 outcomes = ?. The probability of getting one head and one tail from tossing 2 coins is 2 chances out of 4 = ?. The probability of getting zero heads (= 2 tails) is one chance out of 4 = ?.If we continue by tossing n coins we would end up with the number of equivalent outcomes of heads being the same as the same as the number in the nth row of Pascal’s Triangle. For example, if 3 coins are tossed, then the numbers of equivalent outcomes of heads are the numbers 1 3 3 1 which are also the number in 3rd row of Pascal’s Triangle. Sum of The RowsTaking the sum of the numbers on any row of the Pascal’s Triangle and writing the total as power of 2.They will be equal to 2 the nth power or 2n .When n is the number of the row. 20 = 1 21= 1 + 1 = 2 22 = 1 + 2 + 1 = 4 23 = 1 + 3 + 3 + 1 = 8 24 = 1 + 4 + 6 + 4 + 1 =16 METHODOLOGYPascal’s Triangle is a triangular array of number. We have to construct a Pascal’s Triangle. The triangle starts from ‘1’ at the top, which is the 0th row. Numbers are formed by adding the two numbers above them to the left and the right, all number outside. The triangle are considered as ‘0’ 0th row : 1 1st row : 0 + 1=1, 1+0=1 2nd row : 0 + 1=1, 1+1=2, 1+ 0=1 3rd row : 0 + 1=1, 1+2=3, 2+1=3, 1+0=1 4th row : 0 + 1=1, 1+3=4, 3+3=6, 3+1=4, 1+0=1In this way, the rows of the triangle go on infinity. A number in the triangle can also be found by ncr (n chosen) where n is the number of the row and r is the element in that row. The formula for ncr is ncr = n!r!n-r! The factorial symbol ‘!’ means the positive integers multiplied by all the positive integer that are smaller than it Thus 5! = 5*4*3*2*1 = 120 CONCLUSION Focusing on the history and construction of Pascal’s Triangle, as well as on its pattern and applications in several areas of mathematics.. The pattern seen for any row in Pascal's Triangle, the sum of the numbers in that row is 2 raised to the exponent of that row. Pascal’s Triangle also contains the Fibonacci numbers, Isaac Newton, in his "binomial theorem", proved that entries in the Pascal Triangle represent coefficients in the expansion of (a+b)n, where n is any counting number. There are so many patterns and connections involved that is connected to the real world.REFERENCEHazewinkel,Michiel,ed.(2001),?"Pascal triangle",?Encyclopedia of Mathematics,?SpringerGreen, Thomas M and Hamberg, Charles L , Dale Seymour publications, Pascal’s triangle ................
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