This is a part of the series of lectures by Prof V Kannan from University of Hyderabad

1. Module 3: Permutations and
combinations.
Formula for nCr
Modern Mathematics in old
Sanskrit books

2. 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.

3. Three parts of this lecture.
• Part 1. Four quotes from foreign experts.
• Part 2. Four or five Sanskrit passages.
• Part 3. Four remarks.

4. Four foreign experts
• Florian Cajori.
• Mark Dominus.
• W.W.Hunter.
• Severes Sebokht.

5. 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.

6. F.Cajori
• The author of a widely
acclaimed text on the
history of mathematics.
• Florian Cajori
Born 1859
Graubünden, Switzerland
Died 1930 (aged 71)
Berkeley, United States
Occupation : Mathematician

7. About the book
• A History of Mathematics
Book 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

8. Mark Dominus
mjd@plover.com

9. 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.

10. 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.

11. Third Quote
• “The Hindus attained a very high proficiency
in arithmetic and algebra independently of
any foreign influence.”
- W.W.Hunter.

12. Sir William Wilson Hunter
(1840-1900)

13. 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.

14. 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”.

15. Four different countries
• Florian Cajori.
• Mark Dominus.
• W.W.Hunter.
• Severes Sebokht
• Switzerland
• America
• Scotland
• Syria

16. Four different periods
• Florian Cajori.
• Mark Dominus.
• W.W.Hunter.
• Severes Sebokht
• Lived upto 1930
• Lives now.
• Nineteenth century.
• 662 A.D.

17. Four different specializations
• Florian Cajori.
• Mark Dominus.
• W.W.Hunter.
• Severes Sebokht
• Historian
• Computing
• Statistician
• Astronomy

18. 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.

19. Formula for ncr

20. Mahviracharya
• Mahāvīra was a 9th-
century Jain
mathematician from
Mysore, India. He was the
author of
Gaṇitasārasan̄graha,
which revised the
Brāhmasphuṭasiddhānta.
He was patronised by the
Rashtrakuta king
Amoghavarsha. He
separated astrology from
mathematics.

21. 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.
.
.
.

22. Two formulas are same.
• nCr = n!/(r!(n-r)!)
• ncr = n(n-1)…(n-r+1)/1.2….r

23. Lilavati
• स्थानान्तं एकादि चयाङ्क घातः
संख्या विभेिा ननयतकः स्युरङ्कक ः |
• 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.

24. From Brihatsamhita
• पूिेण पूिेण गतेन युक्तं
स्थानं विनान््यं प्रििन्न्त संख्याम ् |
इच्छा विकलपकः क्रमशोz भभनीय
नीते ननिृवतः पुनरन्यनीनतः || 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.

26. 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”.

27. Difficulty
• न गुणो न हरो न कृ नतः
न घनः पृष्टः तथावप िुष्टानाम ् |
गविित गणक बटूनां
स्यात् पातोz िश्यं अङ्कपाशेz न्स्मन् ||
Not multiplication, not division, not squaring,
not cubing, but it is permutation &
combination that certainly pulls down the
wicked and arrogant boys in mathematics.

28. An example to apply the formula
• षणणां चावप रसानां
कषाय नतक्ताम्ल कटुक लिणानाम ् |
मधुररसेन युतानाम ्
भेिान् कथयाधुना गणक ||
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.

29. Working out
• 6C1 + 6C2 + 6C3 + 6C4 + 6C5 + 6C6
= 6 + 15 + 20 +15 + 6 + 1
= 63
= (2^6) – 1.

30. 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.

31. Scene changes to history of these
formulas
• Part 2 ends.
• Seen in part2:
Two verses from Gan. S.S.
One verse from Br.samhita
Two 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?

32. 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.”

33. The previous quote is taken from
Indian Mathematics and Astronomy:
Some Landmarks
By
S Balachandra Rao
Jnana Deep Publications,
Bangalore,
1994,
Pages, VIII + 234,

35. 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)