Lilavati (Bhaskaracharya)

19

The Combinations of Tastes

Verse:
O experts who know the characteristics of mixtures! Tell me, when combinations of one, two, three, and so on are formed from the six tastes original: madhura-kaṭuka-kaṣāyāmla-kṣāra-tikta—sweet, pungent, astringent, sour, alkaline, and bitter—how many distinct varieties of dishes vyañjana: refers here to seasoned dishes or flavor profiles will there be, O mathematician? 22

Numerical Setup:

In the same way, the numbers of combinations for one, two, three, etc., are: 6, 15, 20, 15, 6, 1.

Thus, in the case of a lotus-sacrifice abja-makha: likely a specific ritual arrangement or in a royal palace, the variations of carriers vahana-bheda: possibly refers to transport or arrangements of items total 255 This represents $2^8 - 1$, suggesting a calculation for 8 items. In the second example, the total number of dish variations is 64 This is $2^6$, including the base or "no taste" option; the sum of combinations 6, 15, 20, 15, 6, 1 is 63.

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Arithmetic Series

Rule for Determining Series:
Now, the rule for Series Operations Śreḍhī-vyavahāra: the branch of mathematics dealing with sequences and series:

The sum of a sequence of numbers starting from one is called the Summation Saṅkalita: the sum of the first 'n' natural numbers, calculated as [n × (n+1)] / 2. That summation, when multiplied by the number of terms plus one and divided by three, is called the Sum of Summations Saṅkalitaikya: the sum of triangular numbers, also known as tetrahedral numbers. 1

Problem Statement:
O mathematician, quickly calculate the individual Summations for the numbers one through nine, and then find their Sums of Summations. 2

Setup:
1, 2, 3, 4, 5, 6, 7, 8, 9

The Summations:
1, 3, 6, 10, 15, 21, 28, 36, 45
These are calculated by adding the next number to the previous sum: 1, 1+2=3, 3+3=6, etc.

The Sums of Summations:
1, 4, 10, 20, 35, 56, 84, 120, 165
These are the cumulative totals of the row above: 1, 1+3=4, 4+6=10, etc.