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Contributions of Some Jain Acāryas to Combinatorics 95
That is, when the number of letters - 6, the sūci prastara may be obtained as follows. step 1 step 2 step 3
step 4
step 5 step 6 (fipal)
6
ūra
Thus the sūcī prastāra is 1, 6, 15, 20, 15, 6, 1. Relation between the binomial coefficients and the terms of a binomial expansions.
Acārya Jayadeva's rule, giving the relation between the binomial coefficients and the terms of a binomial expansion, is as given below.
"Starting form the 1st the numbers in the sūci prastara) are the numbers of variations of the metre havirg 0, 1, 2, 3,... ...(and) all, monosyllabic letters, in order."15
Thus, according to the rule, if a metre has n letters, the number ou variations of the metre having r monosyllabic letters (i. e., of the term b.a') is nc, Alternative Rule for sankhyā.
Acārya Jayadeva's alternative rule for finding the total number of variations of a metre is based on a property of binomial coefficients. According to this rule, "the sum of these numbers of sūci prastāra) is the number of variations of the metre.'16 Thus, in symbols, the numbers of variations of a metre having n
letters = nC.
r=0 By the former rule, the number of variations of the metre
= (1+2+4+...+2n-1) + 1 - 2n.
Therefore, nC, 2n, and thus the rule gives another property of the
r=0 binomial coefficients. 15. Jd. viii, 10 (b). The text is,
एकद्वित्रिलघ्नूनि प्रथमाद् गुरुणो भवन्त्येव ॥१०॥ 16. Jd., viii, 11 (b). The text is,
एष्वेव पिण्डितेषु च संख्या प्रस्तार विरचिता भवति ।
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