Bhaskara II Acharya/Lilavati/Chapter XIII/269

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Bhaskara II Acharya: Lilavati Chapter $\text {XIII}$. Combination of Digits: $269$

How many are the variations of the form of the god Sambhu by the exchange of his $10$ attributes held reciprocally in his several hands:
namely, the rope, the elephant's hook, the serpent, the tabor, the skull, the trident, the bedstead, the dagger, the arrow, and the bow:
as those of Hari by the exchange of the mace, the discus, the lotus and the conch?


Solution

For Sambhu: $3 \, 628 \, 800$.

For Hari: $24$.


Proof

We have that Sambhu has $10$ objects which may be held in any of his $10$ hands.

Similarly, Hari has $4$ objects which may be held in any of his $4$ hands.

Thus we are asked: how many different possible representations can be made of each of these two gods, differentiating between them by means of which object is held in which hand?


Hence the question is an instance of Number of Permutations.

The number of permutations of $n$ objects from $n$ is given by:

$^n P_n = n!$

Here we have that Sambhu has $10$ objects, and so:

$10! = 3 \, 628 \, 800$

different representations.

Similarly we have that Hari has $4$ objects, and so:

$4! = 24$

different representations.

$\blacksquare$


Historical Note

The specific names of the gods used in this example vary from text to text.

The canonical rendition from Henry Thomas Colebrooke refers to Sambhu and Hari, while pointing out in a footnote that they are also called Siva and Vishnu.

John Taylor refers to Mahadev and Vishnu.


Vera Sanford points out:

... the Hindus had different names for the god Hari according to the various arrangements of his four attributes in his four hands.

She then goes on to remark:

We may imagine that Bhaskara had this in mind, and thought to astound his readers by the number of names that would have to be invented for Siva had that god been given a name for each of the ways in which his ten attributes could be placed in his ten hands.


Sources

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