PASCAL’S TRIANGLE DEPARTMENT OF ELECTRONICS ; COMMUNICATION ENGINEERING SYNOPSIS Academic Year : 2018 – 2019 Year : IV Sem : VII Sec : A Dept. of ECE, NHCE Page 1PASCAL’S TRIANGLE PASCAL’S TRIANGLE Chapter 1 1.1 Introduction ?In mathematics, Pascal's triangle is a triangular array of binomial coefficients. ?The rows of Pascal's triangle are numbered starting with row 0 at the top (the zero-th row). ?The numbers in each row are numbered from the left beginning With 0.
1.2 Pascal’s triangle(10 rows) Dept. of ECE, NHCE Page 2PASCAL’S TRIANGLE 1.3 Interesting properties ?The diagonals going along the left and right edges contain only 1's. ?The diagonal next to the first diagonal contains all the natural numbers in order starting from 1. ?The next pair of diagonal contain the triangular numbers in order i.
e., 1,3,6 and so on. ?The next pair of diagonals contain the tetrahedral numbers in Order. 1.4 Applications of Pascal’s Triangle ?Binomial expression ?Probability ?Combinations Dept.
of ECE, NHCE Page 3PASCAL’S TRIANGLE Chapter 2 2.1 Fibonacci series 2.2 Binomial expression ?(a+b)^2 = 1a^2 + 2(ab)+ 1b^2 ?The above representation is the coefficient of the expanded Values that follows the Pascal’s triangle according to the power. Dept. of ECE, NHCE Page 4PASCAL’S TRIANGLE 2.3 Probability ?Pascals Triangle can show you how many ways heads and tails can combine. This can be used to find the probability ofany combination.
?In the following slide, H represents Heads and T represents Tails Probability; coin toss example ?For example, if a coin is tossed 4 times, the possible combinations are:- ? HHHH ? HHHT, HHTH, HTHH, THHH ? HHTT, HTHT, HTTH, THHT, THTH, TTHH ? HTTT, THTT, TTHT, TTTH ? TTTT ?From the above we can say that the pattern is 1, 4, 6, 4 1 ?The total number of possibilities can be found by adding all the numbers together. ?Let us take an example of combinations. Dept. of ECE, NHCE Page 5PASCAL’S TRIANGLE ?If there are 5 marbles of different colours, How many different combinations can be made if two marbles are taken out. ? The answer can be found in the 2nd place of row 5, which is 10. This is taking note that the rows start with row 0 and the position in each row also starts with 0.
Dept. of ECE, NHCE Page 6PASCAL’S TRIANGLE Chapter 3 3.1Flowchart Dept. of ECE, NHCE Page 7PASCAL’S TRIANGLE 3.2 Code #include long fun(int y) { int z; long result = 1; for( z = 1 ; z
