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Pascal’s Triangle and it’s application in other subjects and real life.

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A triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)n.  It is named after the 17th-century French mathematician Blaise Pascal.

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It is an equilateral triangle.  It has a triangular array of numbers and its rows are named with 0th row, 1st row, 2nd row, and so on, from the top.  The outer two equal sides of the triangle are filled by 1 at all of the rows. The middle numbers of the triangle are obtained by adding two numbers above it.  By repeating this process we will end up getting a Pascal triangle.   This is a never ending triangle.

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Properties Each number is the sum of the two numbers above it.  The triangle is symmetric.  The diagonals going along the left and right edges contain only 1’s.   The diagonals next to the edge diagonals contain the natural numbers in order.

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The next diagonal is the triangular numbers.  Similarly, the next diagonals are the tetrahedral numbers or triangular pyramidal numbers.

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Sum of the diagonals of the left-justified Pascal Triangle forms a Fibonacci Sequence.  The sums of the rows give the powers of 2.  Each row gives the digits of the powers of 11.  If you shade all the even numbers, you will get a fractal. This is also the recursive of Sierpinski’s Triangle.

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Start with any number in the triangle and       proceed  down the diagonal. Then change the direction in the diagonal for the last number. That last number is the sum of every other number in the diagonal, this is known as Hockey Stick Pattern.  Catalan numbers are found by taking polygons, and finding how many ways they can be partitioned into triangles. These numbers are found in Pascal’s triangle by starting in the 3 row down the middle and subtracting the number adjacent to it.

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Pascal’s triangle determines the coefficients which arise in binomial expansion.  Supposing you have the binomial (a + b) and you want to raise it to a power such as 2 or 3.

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Usage of Pascal’s triangle in real life

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Probability of getting Heads or Tails  Pascal’s triangle can show us the way how heads and tails can combine. This gives the probability of any combination.  H represent heads and T represents tails. HHHH HHHT, HHTH, HTHH, THHH TTHH, HHTT, HTHT, HTTH,THHT, THTH HTTT, THTT, TTHT, TTTH TTTT Thus, the observed pattern is 1,4,6,4,1. (4 th  row of the Pascal’s triangle)

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THE END THANK YOU  DEAR TEACHER  FOR YOUR  ATTENTION

Pascal's Triangle and it's application in other subjects and real life.

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