{"id":3616,"date":"2026-03-21T01:59:59","date_gmt":"2026-03-21T01:59:59","guid":{"rendered":"https:\/\/npk1169-001-site1.mtempurl.com\/?p=3616"},"modified":"2026-04-19T11:19:46","modified_gmt":"2026-04-19T11:19:46","slug":"pascals-triangle-meru-prastara","status":"publish","type":"post","link":"https:\/\/www.bharatwisdom.com\/hi\/mathematics\/pascals-triangle-meru-prastara\/","title":{"rendered":"\u092a\u093e\u0938\u094d\u0915\u0932 \u0915\u093e \u0924\u094d\u0930\u093f\u092d\u0941\u091c (\u092e\u0947\u0930\u0941 \u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0930\u093e) \u2013 \u0938\u0902\u092f\u094b\u091c\u0928\u094b\u0902 \u0915\u0940 \u0938\u0940\u0922\u093c\u0940"},"content":{"rendered":"<div class=\"fusion-fullwidth fullwidth-box fusion-builder-row-1 fusion-flex-container nonhundred-percent-fullwidth non-hundred-percent-height-scrolling\" style=\"--awb-border-radius-top-left:0px;--awb-border-radius-top-right:0px;--awb-border-radius-bottom-right:0px;--awb-border-radius-bottom-left:0px;--awb-flex-wrap:wrap;\" ><div class=\"fusion-builder-row fusion-row fusion-flex-align-items-flex-start fusion-flex-content-wrap\" style=\"max-width:1279.2px;margin-left: calc(-4% \/ 2 );margin-right: calc(-4% \/ 2 );\"><div class=\"fusion-layout-column fusion_builder_column fusion-builder-column-0 fusion_builder_column_1_1 1_1 fusion-flex-column\" style=\"--awb-bg-size:cover;--awb-width-large:100%;--awb-margin-top-large:0px;--awb-spacing-right-large:1.92%;--awb-margin-bottom-large:0px;--awb-spacing-left-large:1.92%;--awb-width-medium:100%;--awb-spacing-right-medium:1.92%;--awb-spacing-left-medium:1.92%;--awb-width-small:100%;--awb-spacing-right-small:1.92%;--awb-spacing-left-small:1.92%;\"><div class=\"fusion-column-wrapper fusion-flex-justify-content-flex-start fusion-content-layout-column\"><div class=\"fusion-text fusion-text-1\"><p id=\"p-rc_ed88986f39805c81-265\" data-path-to-node=\"90\"><span data-path-to-node=\"90,0\"><b data-path-to-node=\"90,0\" data-index-in-node=\"0\">The Concept<\/b> <\/span><span data-path-to-node=\"90,2\"><span class=\"citation-901\">In this triangular array of integers, each number is the sum of the two directly above it<\/span><\/span><span data-path-to-node=\"90,4\">. <\/span><span data-path-to-node=\"90,6\"><span class=\"citation-900\">It is used to calculate binomial coefficients (combinations) and probabilities<\/span><\/span><span data-path-to-node=\"90,8\">.<\/span><\/p>\n<p id=\"p-rc_ed88986f39805c81-266\" data-path-to-node=\"91\"><span data-path-to-node=\"91,0\"><b data-path-to-node=\"91,0\" data-index-in-node=\"0\">The Story<\/b> <\/span><span data-path-to-node=\"91,2\"><span class=\"citation-899\">Long before Blaise Pascal sat down in 17th-century France to map out his famous triangle, Indian poets were using the exact same structure to build a &#8220;Staircase of Mount Meru&#8221; (Meru Prastara)<\/span><\/span><span data-path-to-node=\"91,4\">. <\/span><span data-path-to-node=\"91,6\"><span class=\"citation-898\">In 3000 BCE, the sage Pingala realized that if you wanted to know every possible way to combine short and long syllables in a poem, you needed a pyramid of numbers<\/span><\/span><span data-path-to-node=\"91,8\">. <\/span><span data-path-to-node=\"91,10\"><span class=\"citation-897\">Each row was born from the sum of the two numbers above it<\/span><\/span><span data-path-to-node=\"91,12\">. <\/span><span data-path-to-node=\"91,14\"><span class=\"citation-896\">While the West eventually used this triangle for probability and gambling, India used it to ensure that the rhythm of sacred chants was mathematically perfect<\/span><\/span><span data-path-to-node=\"91,16\">. <\/span><span data-path-to-node=\"91,18\"><span class=\"citation-895\">It wasn\u2019t just a triangle; it was the world\u2019s first combinatorial map<\/span><\/span><span data-path-to-node=\"91,20\">.<\/span><\/p>\n<p data-path-to-node=\"92\"><b data-path-to-node=\"92\" data-index-in-node=\"0\">The Timeline<\/b><\/p>\n<table data-path-to-node=\"93\">\n<thead>\n<tr>\n<td><strong>Milestone<\/strong><\/td>\n<td><strong>Details<\/strong><\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><span data-path-to-node=\"93,1,0,0\"><b data-path-to-node=\"93,1,0,0\" data-index-in-node=\"0\">Western Ref.<\/b><\/span><\/td>\n<td>\n<p id=\"p-rc_ed88986f39805c81-267\" data-path-to-node=\"93,1,1,0\"><span data-path-to-node=\"93,1,1,0,1\"><span class=\"citation-894\">1653 CE (Blaise Pascal) <\/span><\/span><\/p>\n<div class=\"source-inline-chip-container ng-star-inserted\"><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><span data-path-to-node=\"93,2,0,0\"><b data-path-to-node=\"93,2,0,0\" data-index-in-node=\"0\">Indian Source<\/b><\/span><\/td>\n<td>\n<p id=\"p-rc_ed88986f39805c81-268\" data-path-to-node=\"93,2,1,0\"><span data-path-to-node=\"93,2,1,0,0\">Prior to 2,000 BCE (Pingala); <\/span><span data-path-to-node=\"93,2,1,0,2\"><span class=\"citation-893\">10th Century CE (Halayudha commentary) <\/span><\/span><\/p>\n<div class=\"source-inline-chip-container ng-star-inserted\"><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><span data-path-to-node=\"93,3,0,0\"><b data-path-to-node=\"93,3,0,0\" data-index-in-node=\"0\">Chron. Gap<\/b><\/span><\/td>\n<td>\n<p id=\"p-rc_ed88986f39805c81-269\" data-path-to-node=\"93,3,1,0\"><span data-path-to-node=\"93,3,1,0,1\"><span class=\"citation-892\">Over 3,500 Years <\/span><\/span><\/p>\n<div class=\"source-inline-chip-container ng-star-inserted\"><\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p data-path-to-node=\"94\"><b data-path-to-node=\"94\" data-index-in-node=\"0\">The Evidence<\/b><\/p>\n<blockquote data-path-to-node=\"95\">\n<p id=\"p-rc_ed88986f39805c81-270\" data-path-to-node=\"95,0\"><span data-path-to-node=\"95,0,0\"><b data-path-to-node=\"95,0,0\" data-index-in-node=\"0\">Sanskrit Shloka:<\/b> \u092a\u0930\u0947 \u092a\u0942\u0930\u094d\u0923\u092e\u094d \u0964 \u092a\u0930\u0947 \u092a\u0942\u0930\u094d\u0923\u092e\u094d \u0964 \u0909\u092a\u0930\u093f\u0937\u094d\u091f\u093e\u0926\u0947\u0915\u092e\u094d \u0964 <\/span><span data-path-to-node=\"95,0,2\"><span class=\"citation-891\">\u0926\u094d\u0935\u092f\u094b\u0903 \u0915\u094b\u0937\u094d\u0920\u092f\u094b\u0903 \u0938\u092e\u093e\u0939\u093e\u0930\u0903 \u0965 <\/span><\/span><span data-path-to-node=\"95,0,4\"> <b data-path-to-node=\"95,0,4\" data-index-in-node=\"1\">Transliteration:<\/b> Pare p\u016br\u1e47am. Pare p\u016br\u1e47am. Upari\u1e63\u1e6d\u0101dekam. Dvayo\u1e25 ko\u1e63\u1e6dhayo\u1e25 sam\u0101h\u0101ra\u1e25. (Halayudha&#8217;s Commentary) <\/span><span data-path-to-node=\"95,0,7\"> <b data-path-to-node=\"95,0,7\" data-index-in-node=\"1\">Meaning:<\/b> Pingala: &#8220;In the succeeding [row], fill the squares.&#8221; Halayudha (explaining): &#8220;Place one on top. The sum of the two squares above determines the value of the square below.&#8221; (This describes exactly how to generate Pascal&#8217;s Triangle)<\/span><span data-path-to-node=\"95,0,9\"><span class=\"citation-889\">. <\/span><\/span><\/p>\n<div class=\"source-inline-chip-container ng-star-inserted\"><\/div>\n<\/blockquote>\n<p id=\"p-rc_ed88986f39805c81-271\" data-path-to-node=\"96\"><span data-path-to-node=\"96,0\"><b data-path-to-node=\"96,0\" data-index-in-node=\"0\">Related Innovations<\/b> <\/span><span data-path-to-node=\"96,2\"><span class=\"citation-888\">Halayudha used the triangle to calculate the binomial expansion coefficients for <\/span><span class=\"math-inline\" data-math=\"(a+b)^n\" data-index-in-node=\"81\">$(a+b)^n$<\/span><span class=\"citation-888\"> and to solve problems where you had to pick &#8216;k&#8217; items from a list of &#8216;n&#8217;<\/span><\/span><span data-path-to-node=\"96,4\">.<\/span><\/p>\n<p id=\"p-rc_ed88986f39805c81-272\" data-path-to-node=\"97\"><span data-path-to-node=\"97,0\"><b data-path-to-node=\"97,0\" data-index-in-node=\"0\">Fun Fact<\/b> <\/span><span data-path-to-node=\"97,2\"><span class=\"citation-887\">Did you realise that the \u2018stairs\u2019 of the Meru Prastara are the coefficients of <\/span><span class=\"math-inline\" data-math=\"(a+b)^n\" data-index-in-node=\"79\">$(a+b)^n$<\/span><span class=\"citation-887\">, a highly important formula in algebra? <\/span><\/span><\/p>\n<p id=\"p-rc_ed88986f39805c81-273\" data-path-to-node=\"98\"><span data-path-to-node=\"98,0\"><b data-path-to-node=\"98,0\" data-index-in-node=\"0\">The Modern Legacy<\/b> <\/span><span data-path-to-node=\"98,2\"><span class=\"citation-886\">This pattern is the foundation for modern genetics, statistics, and probability theory<\/span><\/span><span data-path-to-node=\"98,4\">.<\/span><\/p>\n<\/div><\/div><\/div><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":3506,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[74],"tags":[544,593,543,592,541,591,542],"class_list":["post-3616","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","tag-ancient-indian-mathematics","tag-binomial-coefficients","tag-combinatorics","tag-halayudha","tag-meru-prastara","tag-pascals-triangle","tag-pingala"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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