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The Author of Ratna Majuşī and the Binomial Expansion
103
Further Advancement Permutation of Digits in which the nunber of digits in a permutation is constant and any digit may be repeated any number of times.
As we have seen earlier, correspondingly, the expansion of varna vșttas is the permutations of the digits ) and 2 in which the number of digits in a particular permutation is constant and any digit may be repeated any number of times.
Nārāyana Pandita (1356 AD) generalises 12 this concept of permutations. All the digits from I uplo q (such that a < 10) take part in his scheme or permutations. The digit q has been named 'the greatest digit' by him. The number of digits in each permutation is constant which has
'the number of places' (-p) by him. Moreover, repetition of digits in such a permutation is permissible.
Rule for Expansion
Nārāyana's rule for the expansion of such permutations may be given as stated below.
“The greatest digit is written in all places (of the first variation). (The next) lesser (digit) is written below the first (available) digit different from unity. Starting froin the left side each time, writing the first available) digit (different from unity) less one' below (that digit, leaving digits on the right as above and writing the greatest digit in each place left of that digit), the process is repeatedly followed until all the p'aces (in an arrangement) are occupied by unity "18
Nārāyaṇa illustrates it his rule with the help of two examples. One of them is as given below. Serial arrangement
when p=3=q. 333 (10) 332
(19) 331 233 (11) 232
(20) 231 133 (12) 132
(21) 131 323 (13) 322
321 223 (14) 222
(23) 221 123 (15) 122
(24) 121 313 (16) 312
(25) 311 213 (17) 212
(26) 211 19) 113 (18) 112
(27) 111
(22)
Saobaoca
12. Singh, Parmanand: Nārāyana's treatment of Net of Numbers,
Ganita Bhārati, 3 (1981), Nos 1-2, p. 23-25. Dvivedi, Padmakara (Ed.): The Ganita Kaumudi by Nārāyana Pandita ( GK.), Princess of Wales, Sarasvati Bhavana Texts,
No. 57, Benares, 1942, Pt. II, xii, 63 64. 14. GK., p. 326-27.
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