Pascal's triangle in Data Structure
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Pascal's Triangle is a triangular array of binomial coefficients named after mathematician Blaise Pascal, although it was studied by others prior to him. It has historical significance, being known in China as early as the 11th century and later presented by Yang Hui, while it has various mathematical properties and applications, including connections to probabilities and number patterns. The triangle reveals intricate patterns, such as the Sierpinski triangle when odd numbers are highlighted.
Pascal's triangle in Data Structure
- 1. Pascal’s Triangle 1 Presented by: The Camouflage Gagan Puri Rabin BK Bikram Bhurtel
- 3. It is a triangular array of the binomial coefficients It is named after a French Mathematician Blaise Pascal Although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy. 3
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- 5. If we expand (X+Y)n then we will get the numbers of n row of Pascal’s triangle Let’s see: 5
- 6. The Pascal’s Triangle is made up of the powers of 11, starting with 110. For nth row 11n Or from the binomial expansion Or trace this pattern: Row 0 6
- 7. Pascal's triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian But it was presented in 13th century by Yang Hui Hence it is still called Yang Hui's triangle in China. The book was written in AD 1303 which is more than 700 years ago and also more than 300 years before Pascal discovered it. Because Pascal collected several results known about the triangle, and employed them to solve problems in probability theory, it was named after him 7
- 8. Pascal’s version Chinese version (yang hui) 8
- 9. The sum of the numbers in any row is equal to 2 to the nth power, where n is the number of the row, i.e., 9 20 = 1 21 = 1+1 = 2 22 = 1+2+1 = 4 23 = 1+3+3+1 = 8 24 = 1+4+6+4+1 = 16
- 10. If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. 10
- 11. 11 1 + 4 = 5 1 + 5 + 15 = 21 1+7+28+84=120
- 12. If you draw a line in the middle of the Pascal’s triangle, the numbers on one side is equal to the numbers on the other side excluding the crossed number. 12
- 13. When we choose a number on the triangle and take 6 other neighboring numbers, it forms a star. Then the product of the numbers pointed by the vertex of a triangle is equal to the product of numbers pointed by the vertex in another triangle. 13 4×210×1 = 7×120×1 840 = 840
- 14. When all the odd numbers in Pascal's Triangle are filled in (black) and the rest are left blank (white), the Sierpinski Triangle is revealed 14
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- 22. References: https://www.slideshare.net/ecooperms/pascals-triangle-slideshow https://www.slideshare.net/rishabhbhndr1/pascal-triangle-36885431 https://www.tutorialspoint.com/learn_c_by_examples/pascals_triangle_progr am_in_c.htm http://britton.disted.camosun.bc.ca/pascal/pascal.html https://www.google.com.np/search?q=pascal+triangles&source=lnms&tbm= isch&sa=X&ved=0ahUKEwiPntvH1avUAhVEQ48KHSOQC3QQ_AUIBig B&biw=1517&bih=698#imgrc=Vnu3EO26ABcNxM: 22
- 23. Queries 23