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Mathematics in Ancient and Medieval India c7.s3: Permutation and Combination

Mathematics in Ancient and Medieval India c7.s3

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Permutation and Combination

c7.s3.1

Smith⁵ opined that "the Hindus seem to have given no attention to permutation and combination until Bhāskara II took it up in his *Līlāvatī*". Chakravarty⁶ has shown that the subject of permutations and combinations is a noteworthy contribution of the Indians and its interest originated before the Christian era. A summary is also given by Srinivasienger.¹ In the early Jain canonical literature, permutation was termed *Vikalpa-gaṇita* and combination *Bhaṅga*. The Jain *Bhagavatī-sūtra* (300 B. C.) calculated the number of combinations of n fundamental categories taken two at a time, three at a time, and so on.² Suśruta (200 B. C.) correctly gave the sum of the combinations of six tastes taken one at a time, two at a time etc. upto all at a time.³ Varāhamihira⁴ (505 A. D.) has stated that "an immense number of perfumes can be made from sixteen substances taken in one, two, three or four proportions and has correctly given the number of perfumes resulting from 16 ingredients (being mixed in all proportions) as 174720". The results given agree with the modern formulae though there is no mention of any such formula in his text.

c7.s3.2

⁵ Smith, D. E. *History of Mathematics*, 2, p. 525, Dover Publication.

c7.s3.3

⁶ Chakravarty, G. 'Growth and Development of Permutations and Combinations in India', *Bulletin of the Calcutta Mathematical Society*, 24, p. 79-88, 1932.

c7.s3.4

¹ Srinivasiengar, C. N. *The History of Ancient Indian Mathematics*, p. 26-28, 73, 83-84, Calcutta, 1967.

c7.s3.5

² Sūtra, 314; vide also *Bulletin of the Calcutta Mathematical Society*, 21, p. 133-136, 1929.

c7.s3.6

³ *Suśrutasaṃhitā*, ch. 63. (*Rasabhedavikalpādhyāya*).

c7.s3.7

⁴ *Bṛhatsaṃhitā*, ch. 77, sl. 13, 14, 17 (for the translation vide Kern's work).

c7.s3.8

The great astronomer and encyclopaedist Varāhamihira applied the same principle in his astrological work *Bṛhajjātaka* in connection with planetary conjunctions.⁵

c7.s3.9

⁵ *Bṛhajjātaka*, rule 3.

c7.s3.10

Mahāvīra⁶ (c. 850 A.D.) first gave the general formula for nc_r. Bhāskara II (c. 1150 A. D.) gave the name *ankapāśa* to the subject of permutation and combination. He also gave the general formulae for nc_r and np_r. Bhāskara II made some other valuable contributions to the subject of permutation and combination. He gave the rules¹ to find :

c7.s3.11

⁶ *Gaṇitasārasaṃgraha*, ch. 3, v. 218.

c7.s3.12

i ) the variation of the number of permutation of a number consisting of n digits of which p number of digits are to be alike and q also are to be alike, the result is (n !)/(p!q!) ;

c7.s3.13

ii ) the variation of numbers each of which contains n digits with a sum equal to S, zero should not be counted as a digit. The result of the problem is given by : S⁻¹c_n-1. This result holds good when 9+n>S.

c7.s3.14

¹ *Līlāvatī*, rules 267, 272, also compare with 274.

c7.s3.15

Bhāskara II gave no proof. Banerjee supplied the plausible proofs to the latter.²

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