Now, in the case of the "net of calculation", the rule of operation for [determining] the different numbers [formed] with specified digits, in one verse:
261. The product of the numbers in the sequence starting with one and ending with the [number of] places is the [number of] different numbers [produced] with digits limited [to that number of places]. [That product] divided by the number of digits, multiplied by the sum of the [specified] digits, [and] added up in [each of the specified] places, is the sum of the amounts [produced with the specified digits].
Here is an example:
262. Say quickly how many different numbers are produced with two and eight, or with three, nine, and eight, and then with [the digits] starting with two and ending with nine, respectively.
Statement: 2, 8. Here there are two places (2); the product of the numbers in the sequence starting with one and ending with the number of places (1, 2) is 2. Therefore there are two (2) different numbers [that may be] produced. Now that product is multiplied by the sum of the digits (10): 20. It is divided by that number of places (2): 10. [This result] is added up in the two [separate decimal] places, [and] the result is the sum of [all the possible] numbers, 110.
Statement in the second example: 3, 9, 8. Here, the product of the numbers in the sequence starting with one (1, 2, 3) is 6. [There are] that many different numbers [produced with those three digits]. Now that product, 6, multiplied by the sum of the digits (20) is 120, [and] divided by the number of digits (3) is 40. [This result] is added up in three places, [and] the result is the sum of the numbers, 4440.
Statement in the third example: 2, 3, 4, 5, 6 7, 8, 9. Here, in the same way, the [number of] different numbers is forty thousand, three hundred and twenty, 40320. And the sum of [those] numbers is twenty-four nikharvas, sixty-three padmas, ninety-nine kotis, ninety-nine laksas, seventy-five thousand, three hundred and sixty: 2463999975360.
Example:
263. How many statues of Sambhu [Siva] [can] there be, with [his attributes] the rope, the elephant-hook, the serpent, the drum, the skull, the trident, the corpse-bier, the dagger, the arrow, and the bow held in his different hands? And how many of Hari [Visnu], with the club, the discus, the lotus, and the conch-shell?
Statement: The [number of] places is 10. The resulting [number of] different statues is 3628800, and in the same way [the number of statues] of Hari is 24.
A different rule for operation, in one verse:
264. The [number of] different [numbers which was explained] previously is divided by the [number of] different [numbers that can be produced] in the [number of] places [occupied by] identical digits---[these divisors are] computed separately [when there is more than one set of identical digits]---[and that] is the [number of] different [numbers produced from the specified digits].
Here is an example:
265. Tell me quickly, mathematician, how many numbers are [produced] with the numbers two, two, one, and one, and [how much is] their sum, and similarly with the numbers four, eight, five, five, and five, if you are skilled in the procedure for the "net of numbers."
Statement: 2, 2, 1, 1. Here, as before, the [number of] different [numbers] is 24. "In the [number of] places [occupied by] identical digits": first, there are in this case two places with identical [digits], [and] as before, there are two different [numbers] produced from two places: 2. And again, there are two [more] places with identical [digits], and so in the same way there are two different [numbers in those places], 2. The previous [number of] different [numbers], 24, is divided by those two [numbers of] different [numbers]; the resulting [number of] different [numbers] is 6, as follows: 2211, 2121, 2112, 1212, 1221, 1122. And the sum of the numbers, [computed] as before, is 9999.
Statement in the second example: 4, 8, 5, 5, 5. Here too, as before, the [number of] different [numbers] is 120. [That is] divided by the [number of] different [numbers] produced from three places (6). The result is 20, as follows: 48555, 84555, 54855, 58455, 55485, 55845, 55548, 55584, 45855, 45585, 45558, 85455, 85545, 85554, 54585, 58545, 55458, 55854, 54558, 58554, that is, twenty [numbers]. And now the sum of the numbers is 1199988.
The rule of operation in permutations with distinct and unspecified digits, in half a verse:
266. And the product of the [sequence of] numbers decreasing by one from the last [i.e. 9] and ending with [the number of] places is the [number of] different numbers [produced] with distinct digits.
Example:
267. Tell [me] if you know how many different numbers are [produced] with any digits but zero interchanged with each other in six places.
Here the "last number" is nine, 9. Setting [those] down decreased by one in six places, the statement is 9, 8, 7, 6, 5, 4. When the product of those [is computed], the resulting [number of] different numbers is 60480.
Another rule of operation, in two verses:
268. If the sum of the digits is specified, this [sequence of numbers] decreasing by one from one less than the sum of the digits down to one less than the [number of] places is divided by [an increasing sequence] beginning with one. The total product of those [quotients] is the [number of] different numbers.
269. [This rule] is understood [as] stated [only] for the case when the sum of the digits is less than the number of places plus nine. [Only] a summary is stated [here] for fear of [going on too] long, since the ocean of calculation is endless.
Example:
270. Tell [me] if you know how many different numbers there are with five digits, the sum of which is thirteen.
Here the sum of the digits is 13, minus one, 12. The [sequence] decreasing by one from that to one less the number of digits is divided by [an increasing sequence] beginning with one: the result is 12/1, 11/2, 10/3, 9/4. The total of those multiplied together is the [number of] different numbers, 495.
271. Although [neither] the multiplier nor divisor nor square nor cube is asked [for], bad [students, although] conceited [about their abilities as] mathematicians, [will] certainly [make] a mistake in [calculating about] this "net of numbers."
That is the "net of numbers" in the Lilavati.
272. Those who hold at their throats the accurate Lilavati illustrating elegant sentences, [whose] parts are adorned with excellent [rules for] reduction and multiplication and squaring [etc.], attain ever-increasing happiness and success.