Transcript
Module 3: Permutations and combinations.
Formula for nCr
Modern Mathematics in old Sanskrit books
The formula
• nCr = n!/(r!(n-r)!)• nCr is the number of ways of choosing r
objects from n objects.• N! is the product of all integers from 1 to N.
Three parts of this lecture.
• Part 1. Four quotes from foreign experts.• Part 2. Four or five Sanskrit passages.• Part 3. Four remarks.
Four foreign experts
• Florian Cajori.• Mark Dominus.• W.W.Hunter.• Severes Sebokht.
First Quote
• “The Hindus solved problems in interest, discount, partnership, alligation, summation of arithmetical and geometric series, and devised rules for determining the numbers of combinations and permutations. It may be added here that chess, the profoundest of all games, had its origin in India.
- F.Cajori.
F.Cajori
• The author of a widely acclaimed text on the history of mathematics.
• Florian Cajori
Born 1859Graubünden, Switzerland
Died 1930 (aged 71)Berkeley, United States
Occupation : Mathematician
About the book
• A History of MathematicsBook by Florian Cajori• Many of the earliest
books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive.
• Published: 1893• Author: Florian Cajori
Second Quote
• “I ran across the algorithm last year while I was reading the Lilavati, a treatise on arithmetic, written about 850 years ago in India… This algorithm is simple, ancient, efficient and convenient…. Why isn’t this better known?”
- Mark Jason Dominus.
Main idea of the algorithm
• (n+1)C(k+1) = (n+1)(nCk)/(k+1).• The University of discourse• www.prover.com• How to calculate binomial coefficients.
Third Quote
• “The Hindus attained a very high proficiency in arithmetic and algebra independently of any foreign influence.”
- W.W.Hunter.
Sir William Wilson Hunter (1840-1900)
Severes Sebokht
• Seboukt of Nisibis. was a Syrian scholar and bishop who was born in Syria in 575 and died in 667. A member of Syrian orthodox church.
• His major legacy is the transmission of the Indian number system to the Islamic world. He was perhaps the first Syrian to mention the Indian number system.
Fourth Quote
• “I will omit all the discussion of Indians, people not the same as Syrians, of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; calculations which surpass description”.
Four different countries
• Florian Cajori.• Mark Dominus.• W.W.Hunter.• Severes Sebokht
• Switzerland• America• Scotland• Syria
Four different periods
• Florian Cajori.• Mark Dominus.• W.W.Hunter.• Severes Sebokht
• Lived upto 1930• Lives now.• Nineteenth century.• 662 A.D.
Four different specializations
• Florian Cajori.• Mark Dominus.• W.W.Hunter.• Severes Sebokht
• Historian• Computing• Statistician• Astronomy
Change scene to India
• Part 1 ends• Four quotes of
appreciation have been seen so far.
• Two on the nCr formula. Two on arithmetics and algebra in general.
• Part 2 starts• Four or five Samskruta
slokas will be seen. • All these will be on
permutations and combinations.
Formula for ncr
Mahviracharya
• Mahāvīra was a 9th-century Jain mathematician from Mysore, India. He was the author of Gaṇitasārasa graha, which n̄�revised the Brāhmasphuṭasiddhānta. He was patronised by the Rashtrakuta king Amoghavarsha. He separated astrology from mathematics.
Example
• 10 C 4=?• 10.9.8.7/1.2.3.4 = 5040/24 = 210.• 1,2,3,4. 2,3,4,5. 1,3,5,7. . . .
Two formulas are same.
• nCr = n!/(r!(n-r)!)• ncr = n(n-1)…(n-r+1)/1.2….r
Lilavati
• स्था�ना�न्तं� एका�दि� चया�ङ्� का घा�तं� सं�ख्या� वि�भे��� विनायातं�� स्या�रङ्� का� � |• If n (<10) is the number of digits, take the product
of all terms in the progression (of integers) starting from 1 ending with n. This gives the total number of integers with n digits, where digits are distinct elements of a given n-subset of {1,2,…9}.
• Eg. How many four digit numbers are there where the digits are 2,3,7,9? Answer:24.
From Brihatsamhita
• पू���ण पू���ण गतं�ना या�क्तं� स्था�ना� वि�ना�न्त्या� प्र��न्तिन्तं सं�ख्या�म्� | इच्छा� वि�काल्पू�� क्रम्शो+z भिभेना-या ना-तं� विना�.भि/� पू�नारन्याना-वितं� || B.S.77-22.Write the numbers 1,2,3,…,n in the direct order and
in reverse order below and retain only according to the requirement (as desired). The result is ncr = n(n-1)…(n-r+1)/1.2….r
- Translation by Prof.S.Madhavan.
Brihat-Samhita
An important contribution of Varahamihira is the encyclopedic Brihat-Samhita. It covers wide ranging subjects of human interest, including astrology, planetary movements, eclipses, rainfall, clouds, architecture, growth of crops, manufacture of perfume, matrimony, domestic relations, gems, pearls, and rituals. The volume expounds on gemstone evaluation criterion found in the Garuda Purana, and elaborates on the sacred Nine Pearls from the same text. It contains 106 chapters and is known as the "great compilation”.
Difficulty
• ना ग�ण+ ना हर+ ना का. वितं� ना घाना� पू.ष्टः� तंथा�विपू दुष्टः�ना�म्� | गर्वि�5तं गणका बटू�ना�� स्या�तं� पू�तं+z �श्या� अङ्� कापू�शो�z स्मिस्म्ना� ||Not multiplication, not division, not squaring,
not cubing, but it is permutation & combination that certainly pulls down the wicked and arrogant boys in mathematics.
An example to apply the formula
• षण्ण�� च�विपू रसं�ना�� काष�या वितंक्तं�म्ल काटू�का ल�ण�ना�म्� | म्धु�ररसं�ना या�तं�ना�म्� भे���ना� काथाया�धु�ना� गणका ||O mathematician! How many taste-
combinations are there, formed from one or more of the six tastes, bitter, pungent, sour, hot, saline and sweet? Now tell me.
Working out
• 6C1 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6 = 6 + 15 + 20 +15 + 6 + 1 = 63 = (2^6) – 1.
Reach of Mahavira’s book
• 870 A.D. Sanskrit book written by Mahavira.• 1050 A.D. Telugu version by Pavaluri Mallana.• 1912 A.D. English Translation by Rangacharya• 1963 A.D. Hindi translation by L.C.Jain.• 2000 A.D.Kannadatranslation, Padmavatamma• 2003 A.D. Telugu translation, Telugu Academy.
Scene changes to history of these formulas
• Part 2 ends.• Seen in part2: Two verses from Gan. S.S.One verse from Br.samhitaTwo verses from Lilavati.
• Part 3 starts.• Four questions:Do all historians agree?Who discovered it first?How was it in pre-
Christian era?Were there improved
formulae later in Sanskrit?
Quote
• “The general formula for nCr has wrongly been attributed to Herigone by Prof.D.E.Smith in his History of Mathematics published in 1925. Ironically, thirteen years earlier, Prof.Smith had written a foreword to Mahavira’s Ganitasarsangraha published in 1912. The moral of the story is that a historian, especially of science, should have an unbiased mind, as also a thorough consistency in his presentations and assessments.”
The previous quote is taken from
Indian Mathematics and Astronomy: Some Landmarks By S Balachandra Rao Jnana Deep Publications, Bangalore, 1994, Pages, VIII + 234,
Who discovered the formula for nCr first?
• Mahaviracharya in 9th century? (He may be the first to state it so clearly.) “The credit of giving out this general formula for nCr for the first time in the history of world mathematics goes uniquely to Mahavira”- Dr.S.Balachandra Rao.
• Varahamihira in 5th & 6th century? (May be.)• Known in India still earlier? (May be. Certainly
some particular cases were known in 200B.C)
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